Chapter 12 Areas Related to Circles (Additional Questions)

The perimeter of a circle is also known as its Circumference.

Therefore, the correct option is (C).

The formula for the circumference of a circle with radius $r$ is $2\pi r$. This matches option (B).

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The diameter $d$ of a circle is twice its radius $r$, i.e., $d = 2r$.

Substituting $2r$ with $d$ in the formula $2\pi r$, we get $\pi(2r) = \pi d$. So, $\pi d$ is also a formula for the circumference, where $d$ is the diameter. This matches option (C).

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Since both $2\pi r$ and $\pi d$ are correct formulas for the circumference of a circle, options (B) and (C) are both correct.

Therefore, the correct option is (D) Both (B) and (C).

The formula for the area of a circle with radius $r$ is $\pi r^2$.

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Comparing this with the given options, we find that option (B) matches the correct formula.

Option (A) is the formula for the circumference of a circle.

Option (C) is incorrect; the area in terms of diameter $d$ is $\pi (\frac{d}{2})^2 = \frac{1}{4}\pi d^2$.

Option (D) is the formula for the area of a semicircle.

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Therefore, the correct option is (B) $\pi r^2$.

Given:

Circumference of the circle, $C = 44 \text{ cm}$.

Value of $\pi = \frac{22}{7}$.

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To Find:

The radius of the circle, $r$.

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Solution:

The formula for the circumference of a circle is $C = 2\pi r$, where $r$ is the radius.

Substitute the given values into the formula:

Simplify the equation:

$44 = \frac{44}{7} \times r$

To find $r$, multiply both sides by $\frac{7}{44}$:

Cancel out the common factor 44:

$r = 7$

The radius of the circle is $7 \text{ cm}$.

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Comparing this result with the given options, we find that it matches option (A).

Therefore, the correct option is (A) 7 cm.

Given:

Diameter of the circle, $d = 14 \text{ cm}$.

Value of $\pi = \frac{22}{7}$.

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To Find:

The area of the circle.

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Solution:

The radius of the circle, $r$, is half of the diameter.

Substitute the given diameter:

$r = \frac{14}{2} \text{ cm}$

$r = 7 \text{ cm}$

The formula for the area of a circle is $A = \pi r^2$, where $r$ is the radius.

Substitute the values of $\pi$ and $r$ into the formula:

Calculate the square of the radius:

Area $= \frac{22}{7} \times 49 \text{ cm}^2$

Simplify the expression:

Cancel the common factor 7:

Area $= 22 \times 7 \text{ cm}^2$

Area $= 154 \text{ cm}^2$

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The area of the circle is $154 \text{ cm}^2$.

Comparing this result with the given options, we find that it matches option (A).

Therefore, the correct option is (A) $154 \text{ cm}^2$.

A sector of a circle is the portion of a circle enclosed by two radii and the arc connecting their endpoints.

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Option (A) describes a segment of a circle.

Option (C) describes a semicircle, which is a specific type of segment (where the chord is a diameter) and also a specific type of sector (where the central angle is $180^\circ$). However, the general definition of a sector is broader.

Option (D) does not define a standard region of a circle.

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Therefore, the correct description of a sector of a circle is the region between two radii and an arc.

The correct option is (B) Two radii and an arc.

The area of a sector of a circle is a fraction of the total area of the circle. The fraction is determined by the ratio of the central angle of the sector to the total angle of a circle ($360^\circ$).

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The formula for the area of a circle with radius $r$ is $A_{circle} = \pi r^2$.

The formula for the area of a sector with radius $r$ and central angle $\theta$ (in degrees) is given by:

Substitute the formula for the area of the circle into equation (i):

Area of sector $= \frac{\theta}{360} \times \pi r^2$

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Comparing this formula with the given options:

Option (A) $\frac{\theta}{180} \times \pi r^2$ is incorrect.

Option (B) $\frac{\theta}{360} \times 2\pi r$ is the formula for the length of the arc of the sector, not its area.

Option (C) $\frac{\theta}{360} \times \pi r^2$ matches the derived formula for the area of the sector.

Option (D) $\frac{\theta}{180} \times 2\pi r$ is incorrect.

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Therefore, the correct option is (C) $\frac{\theta}{360} \times \pi r^2$.

Given:

Radius of the sector, $r = 6 \text{ cm}$.

Central angle of the sector, $\theta = 60^\circ$.

Value of $\pi = \frac{22}{7}$.

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To Find:

The area of the sector.

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Solution:

The formula for the area of a sector with radius $r$ and central angle $\theta$ (in degrees) is:

Area of sector $= \frac{\theta}{360^\circ} \times \pi r^2$

Substitute the given values into the formula:

Simplify the fraction $\frac{60}{360}$:

Calculate $(6 \text{ cm})^2$:

$(6 \text{ cm})^2 = 36 \text{ cm}^2$

Substitute these values back into equation (i):

Simplify the expression:

Area $= \frac{22}{7} \times 6 \text{ cm}^2$

Area $= \frac{22 \times 6}{7} \text{ cm}^2$

Area $= \frac{132}{7} \text{ cm}^2$

Now, calculate the decimal value:

$\frac{132}{7} \approx 18.85714...$

Rounding to two decimal places, we get approximately $18.86 \text{ cm}^2$. However, looking at the options, option (A) is $18.85 \text{ cm}^2$ (approx.). Let's recheck the options and calculation.

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Comparing the result with the options:

(A) $18.85 \text{ cm}^2$ (approx.)

(B) $37.71 \text{ cm}^2$ (approx.) - This is roughly $2 \times 18.85$. It could be related to circumference or area with double the angle/radius in some way, but not the sector area here.

(C) $113.14 \text{ cm}^2$ (approx.) - This is roughly $6 \times 18.85$. The area of the full circle would be $\frac{22}{7} \times 6^2 = \frac{22}{7} \times 36 = \frac{792}{7} \approx 113.14 \text{ cm}^2$. This is the area of the whole circle.

(D) $36 \text{ cm}^2$ - This is $r^2$.

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Our calculated value $\frac{132}{7} \approx 18.857...$ is closest to $18.85 \text{ cm}^2$. The term "(approx.)" in the option suggests that rounding is expected.

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Therefore, the correct option is (A) $18.85 \text{ cm}^2$ (approx.).

A segment of a circle is the region bounded by a chord and the arc subtended by the chord.

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Comparing this definition with the given options:

Option (A) "A chord and an arc" correctly describes a segment of a circle.

Option (B) "Two radii and an arc" describes a sector of a circle.

Option (C) "A diameter and a semicircle" describes a semicircle, which is a special case of a segment (where the chord is a diameter) and also a special case of a sector (where the central angle is $180^\circ$). However, the general definition of a segment is the region between any chord and its corresponding arc.

Option (D) "Two parallel chords" does not define a standard region of a circle with a specific name.

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Therefore, the correct description of a segment of a circle is the region between a chord and an arc.

The correct option is (A) A chord and an arc.

A segment of a circle is the region bounded by a chord and its corresponding arc.

When we consider the sector corresponding to this arc, it is bounded by the two radii to the endpoints of the arc and the arc itself.

The chord connecting the endpoints of the arc forms a triangle with the two radii.

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The area of the sector includes the area of this triangle and the area of the segment.

Therefore, the area of the segment can be obtained by subtracting the area of the triangle from the area of the sector.

The problem provides that the area of the triangle is $\frac{1}{2}r^2 \sin \theta$, and the area of the sector is $\frac{\theta}{360} \pi r^2$ (for $\theta$ in degrees) or $\frac{1}{2}r^2 \theta$ (for $\theta$ in radians).

Substituting these into equation (i):

Area of segment $= \frac{\theta}{360} \pi r^2 - \frac{1}{2}r^2 \sin \theta$ (if $\theta$ is in degrees)

or

Area of segment $= \frac{1}{2}r^2 \theta - \frac{1}{2}r^2 \sin \theta$ (if $\theta$ is in radians)

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Comparing the relationship in equation (i) with the given options, we find that option (A) directly states this relationship.

Option (A) is Area of sector - Area of triangle.

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Therefore, the correct option is (A) Area of sector - Area of triangle.

Given:

Perimeter of a circle is equal to the perimeter of a square.

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To Find:

The ratio of their areas (Area of Circle : Area of Square).

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Solution:

Let the radius of the circle be $r$ and the side of the square be $s$.

The perimeter of the circle is $2\pi r$.

The perimeter of the square is $4s$.

According to the given information, the perimeters are equal:

We can express $s$ in terms of $r$ from this equation:

Simplify equation (i):

$s = \frac{\pi r}{2}$

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Now, let's find the areas of the circle and the square.

The area of the circle is $A_{circle} = \pi r^2$.

The area of the square is $A_{square} = s^2$.

Substitute the expression for $s$ from equation (i) into the formula for the area of the square:

$A_{square} = \frac{\pi^2 r^2}{4}$

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Now, we need to find the ratio of the area of the circle to the area of the square.

Simplify the expression for the ratio:

Cancel out the common terms $r^2$ and one power of $\pi$:

Ratio $= \frac{4}{\pi}$

The ratio of the area of the circle to the area of the square is $4:\pi$.

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Comparing this result with the given options, we find that it matches option (C).

Therefore, the correct option is (C) $4:\pi$.

Given:

Length of the minute hand = Radius of the circle, $r = 14 \text{ cm}$.

Time duration $= 5 \text{ minutes}$.

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To Find:

The area swept by the minute hand in 5 minutes.

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Solution:

The minute hand completes a full circle ($360^\circ$) in 60 minutes.

The angle swept by the minute hand in 1 minute is:

Angle per minute $= 6^\circ$ per minute.

The angle swept by the minute hand in 5 minutes is:

$\theta = 30^\circ$

The area swept by the minute hand in 5 minutes is the area of a sector with radius $r = 14 \text{ cm}$ and central angle $\theta = 30^\circ$.

The formula for the area of a sector is: Area $= \frac{\theta}{360^\circ} \times \pi r^2$.

Let's use $\pi = \frac{22}{7}$ as it is commonly used in such problems with radii being multiples of 7, and options are given in specific values.

Substitute the given values into the formula:

Simplify the fraction $\frac{30}{360}$:

Calculate $(14 \text{ cm})^2$:

$(14 \text{ cm})^2 = 196 \text{ cm}^2$

Substitute these values back into equation (ii):

Simplify the expression:

Area $= \frac{22 \times 196}{12 \times 7} \text{ cm}^2$

Cancel out common factors. 196 is divisible by 7 ($196 = 7 \times 28$). 12 and 22 are divisible by 2. 12 and 28 are divisible by 4.

Area $= \frac{11 \times 14}{3} \text{ cm}^2$

Area $= \frac{154}{3} \text{ cm}^2$

Calculate the decimal value:

$\frac{154}{3} \approx 51.333... \text{ cm}^2$

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Comparing this result with the given options, we find that it approximately matches option (D).

Therefore, the correct option is (D) $51.33 \text{ cm}^2$ (approx.).

Analysis of Assertion (A):

Assertion (A) states that the area of a sector with angle $90^\circ$ is one-fourth the area of the circle.

The formula for the area of a sector with radius $r$ and central angle $\theta$ (in degrees) is given by:

Area of sector $= \frac{\theta}{360^\circ} \times \pi r^2$

Given $\theta = 90^\circ$ and the area of the circle is $\pi r^2$.

Substitute $\theta = 90^\circ$ into the sector area formula:

Simplify the fraction:

So, from equation (i):

Area of sector $= \frac{1}{4} \times \pi r^2$

This is indeed one-fourth the area of the circle ($\pi r^2$).

Thus, Assertion (A) is True.

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Analysis of Reason (R):

Reason (R) states that a sector with angle $90^\circ$ represents a quarter of the circle ($90/360 = 1/4$).

The fraction of a circle that a sector represents is determined by the ratio of its central angle to the total angle of a circle ($360^\circ$).

For a sector with central angle $90^\circ$, the ratio is:

Simplify the ratio from equation (ii):

Ratio $= \frac{1}{4}$

This means a sector with a $90^\circ$ angle occupies one-fourth of the total angular space of the circle.

Thus, Reason (R) is True.

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Conclusion:

Both Assertion (A) and Reason (R) are true.

The area of a sector is directly proportional to its central angle. The formula for the area of a sector arises from taking the fraction of the circle's area that corresponds to the fraction of the total angle. Reason (R) correctly identifies the fraction of the circle represented by the $90^\circ$ angle, which is exactly why the area of the sector is that fraction of the total area.

Therefore, Reason (R) is the correct explanation for Assertion (A).

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Based on the analysis, both A and R are true and R is the correct explanation of A.

The correct option is (A) Both A and R are true and R is the correct explanation of A.

Analysis of Assertion (A):

Assertion (A) states that if the circumference of a circle is numerically equal to its area, then the radius is 2 units.

Let the radius of the circle be $r$.

Circumference of the circle, $C = 2\pi r$.

Area of the circle, $A = \pi r^2$.

According to the condition in Assertion (A), the circumference is numerically equal to the area:

To find the value of $r$, we can rearrange equation (i):

$\pi r^2 - 2\pi r = 0$

Factor out $\pi r$:

$\pi r (r - 2) = 0$

This equation holds true if $\pi r = 0$ or $r - 2 = 0$.

Since $\pi$ is a constant and $r$ is a radius (which must be positive, $r > 0$), $\pi r$ cannot be 0.

Therefore, we must have $r - 2 = 0$.

So, the radius of the circle is 2 units when its circumference is numerically equal to its area.

Thus, Assertion (A) is True.

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Analysis of Reason (R):

Reason (R) states that $2\pi r = \pi r^2$ implies $r=2$ (for $r>0$).

As shown in the analysis of Assertion (A), the equation $2\pi r = \pi r^2$, when solved for $r$ under the condition $r > 0$, leads to $r = 2$.

The steps involved are:

Since $r > 0$, we can divide both sides of equation (ii) by $\pi r$:

$2 = r$

Thus, Reason (R) is True.

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Conclusion:

Both Assertion (A) and Reason (R) are true. Reason (R) provides the exact mathematical derivation that proves the relationship stated in Assertion (A). Therefore, Reason (R) is the correct explanation for Assertion (A).

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Based on the analysis, both A and R are true and R is the correct explanation of A.

The correct option is (A) Both A and R are true and R is the correct explanation of A.

Let's match each term from Column A with its corresponding formula in Column B.

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(i) Circumference of circle: The formula for the circumference of a circle with radius $r$ is $2\pi r$. This matches formula (b).

(i) $\to$ (b)

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(ii) Area of circle: The formula for the area of a circle with radius $r$ is $\pi r^2$. This matches formula (c).

(ii) $\to$ (c)

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(iii) Length of arc of sector (angle $\theta$): The length of an arc of a sector with radius $r$ and central angle $\theta$ (in degrees) is a fraction of the circle's circumference, given by $\frac{\theta}{360} \times 2\pi r$. This matches formula (d).

(iii) $\to$ (d)

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(iv) Area of sector (angle $\theta$): The area of a sector with radius $r$ and central angle $\theta$ (in degrees) is a fraction of the circle's area, given by $\frac{\theta}{360} \times \pi r^2$. This matches formula (a).

(iv) $\to$ (a)

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The correct matching is:

• (i) - (b)
• (ii) - (c)
• (iii) - (d)
• (iv) - (a)

Comparing this matching with the given options, we find that option (A) corresponds to this combination.

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Therefore, the correct option is (A) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a).

Given:

Radius of the circular cardboard, $r = 14 \text{ cm}$.

Angle of the sector cut out, $\theta = 90^\circ$.

We will use $\pi = \frac{22}{7}$ for calculations involving radius 14.

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To Find:

The area of the piece cut out (which is the area of the sector).

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Solution:

The area of a sector of a circle with radius $r$ and central angle $\theta$ (in degrees) is given by the formula:

Area of sector $= \frac{\theta}{360^\circ} \times \pi r^2$

Substitute the given values into the formula:

Simplify the fraction $\frac{90}{360}$:

Calculate $(14 \text{ cm})^2$:

$(14 \text{ cm})^2 = 196 \text{ cm}^2$

Substitute these values back into equation (i):

Simplify the expression:

Cancel out common factors. 196 is divisible by 7 ($196 = 7 \times 28$). 4 divides 28.

Area $= 22 \times 7 \text{ cm}^2$

Area $= 154 \text{ cm}^2$

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The area of the piece cut out is $154 \text{ cm}^2$.

Comparing this result with the given options, we find that it matches option (A).

Therefore, the correct option is (A) $154 \text{ cm}^2$.

Given:

Radius of the circular cardboard, $r = 14 \text{ cm}$.

Angle of the sector cut out, $\theta_{cut} = 90^\circ$.

From the previous question (Question 16), the area of the sector cut out is $154 \text{ cm}^2$.

We use $\pi = \frac{22}{7}$.

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To Find:

The area of the remaining piece of cardboard.

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Solution:

The remaining piece of cardboard is the area of the full circle minus the area of the sector cut out.

First, calculate the area of the full circle with radius $r = 14 \text{ cm}$.

Area of circle $= \pi r^2$

Calculate $(14 \text{ cm})^2$:

$(14 \text{ cm})^2 = 196 \text{ cm}^2$

Substitute this value back into equation (i):

Simplify the expression:

Cancel out the common factor 7:

Area of circle $= 22 \times 28 \text{ cm}^2$

Area of circle $= 616 \text{ cm}^2$

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The area of the sector cut out is $154 \text{ cm}^2$ (as calculated in Question 16).

Area of remaining piece = Area of full circle - Area of sector cut out

Perform the subtraction:

$\begin{array}{cc} & 6 & 1 & 6 \\ - & 1 & 5 & 4 \\ \hline & 4 & 6 & 2 \\ \hline \end{array}$

Area of remaining piece $= 462 \text{ cm}^2$

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The area of the remaining piece of cardboard is $462 \text{ cm}^2$.

Comparing this result with the given options, we find that it matches option (A).

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Alternatively:

The remaining piece is a major sector with a central angle of $360^\circ - 90^\circ = 270^\circ$.

Area of remaining sector $= \frac{\theta_{remaining}}{360^\circ} \times \pi r^2$

Simplify the fraction $\frac{270}{360} = \frac{3}{4}$.

Area $= \frac{3}{4} \times \frac{22}{7} \times 196 \text{ cm}^2$

Area $= \frac{3}{4} \times 22 \times 28 \text{ cm}^2$

Area $= 3 \times 22 \times \frac{28}{4} \text{ cm}^2$

Area $= 3 \times 22 \times 7 \text{ cm}^2$

Area $= 66 \times 7 \text{ cm}^2$

Area $= 462 \text{ cm}^2$

Both methods yield the same result.

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Therefore, the correct option is (A) $462 \text{ cm}^2$.

Given:

Radius of the sector, $r = 10 \text{ cm}$.

Central angle of the sector, $\theta = 180^\circ$.

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To Find:

The length of the arc of the sector.

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Solution:

The formula for the length of an arc ($L$) of a sector with radius $r$ and central angle $\theta$ (in degrees) is:

Substitute the given values into the formula (i):

Simplify the fraction $\frac{180}{360}$:

Substitute this back into the arc length calculation:

Length of arc $= \frac{1}{2} \times 2\pi \times 10 \text{ cm}$

Cancel out the factor of 2:

Length of arc $= \cancel{\frac{1}{2}} \times \cancel{2}\pi \times 10 \text{ cm}$

Length of arc $= \pi \times 10 \text{ cm}$

Length of arc $= 10\pi \text{ cm}$

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The length of the arc of the sector is $10\pi \text{ cm}$.

Comparing this result with the given options, we find that it matches option (A).

Therefore, the correct option is (A) $10\pi$ cm.

Given:

Radius of the semicircle, $r = 7 \text{ cm}$.

Value of $\pi = \frac{22}{7}$.

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To Find:

The area of the semicircle.

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Solution:

A semicircle is exactly half of a circle. Therefore, its area is half the area of the full circle with the same radius.

The formula for the area of a circle with radius $r$ is $A_{circle} = \pi r^2$.

The formula for the area of a semicircle is:

Substitute the given values into the formula (i):

Calculate $(7 \text{ cm})^2$:

$(7 \text{ cm})^2 = 49 \text{ cm}^2$

Substitute this value back into the area calculation:

Simplify the expression from equation (ii):

Area $= \frac{22 \times 49}{2 \times 7} \text{ cm}^2$

Cancel out common factors. 22 and 2 are divisible by 2. 49 and 7 are divisible by 7.

Area $= 11 \times 7 \text{ cm}^2$

Area $= 77 \text{ cm}^2$

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The area of the semicircle is $77 \text{ cm}^2$.

Comparing this result with the given options, we find that it matches option (B).

Therefore, the correct option is (B) $77 \text{ cm}^2$.

Given:

Area of the circle, $A = 154 \text{ cm}^2$.

We will use $\pi = \frac{22}{7}$.

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To Find:

The radius of the circle, $r$.

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Solution:

The formula for the area of a circle is $A = \pi r^2$, where $r$ is the radius.

Substitute the given area and the value of $\pi$ into the formula:

To find $r^2$, multiply both sides of equation (i) by $\frac{7}{22}$:

Simplify the expression:

$r^2 = 7 \times 7$

$r^2 = 49$

Take the square root of both sides to find $r$:

Since the radius must be a positive value:

The radius of the circle is $7 \text{ cm}$.

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Comparing this result with the given options, we find that it matches option (A).

Therefore, the correct option is (A) 7 cm.

Given:

A circle with radius $r$ and diameter $d$.

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To Find:

The ratio of its circumference to its diameter.

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Solution:

The formula for the circumference of a circle with radius $r$ is $C = 2\pi r$.

The diameter $d$ of a circle is twice its radius $r$, i.e., $d = 2r$.

The ratio of the circumference to the diameter is given by:

Substitute the formulas for circumference and diameter into equation (i):

Cancel out the common term $2r$ from the numerator and the denominator:

Ratio $= \pi$

The ratio of the circumference of a circle to its diameter is always the constant $\pi$.

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Comparing this result with the given options, we find that it matches option (A).

Therefore, the correct option is (A) $\pi$.

Given:

Radius of the first circle, $r_1 = 8 \text{ cm}$.

Radius of the second circle, $r_2 = 6 \text{ cm}$.

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To Find:

The radius ($R$) of a new circle whose area is equal to the sum of the areas of the two given circles.

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Solution:

The area of the first circle is $A_1 = \pi r_1^2$.

$A_1 = \pi (8 \text{ cm})^2 = 64\pi \text{ cm}^2$

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The area of the second circle is $A_2 = \pi r_2^2$.

$A_2 = \pi (6 \text{ cm})^2 = 36\pi \text{ cm}^2$

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Let the radius of the new circle be $R$. Its area is $A_{new} = \pi R^2$.

According to the problem, the area of the new circle is the sum of the areas of the two given circles:

Substitute the formulas for the areas into equation (i):

Factor out $\pi$ from the right side:

$\pi R^2 = \pi (r_1^2 + r_2^2)$

Divide both sides by $\pi$ (since $\pi \neq 0$):

Substitute the given values of $r_1$ and $r_2$:

Calculate the squares:

$R^2 = 64 \text{ cm}^2 + 36 \text{ cm}^2$

$R^2 = 100 \text{ cm}^2$

Take the square root of both sides to find $R$:

Since radius must be positive:

$R = 10 \text{ cm}$

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The radius of the new circle is $10 \text{ cm}$.

Comparing this result with the given options, we find that it matches option (B).

Therefore, the correct option is (B) 10 cm.

Given:

Radius of the circle $= r$.

A square is inscribed in the circle such that its vertices lie on the circle.

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To Find:

The area of the largest square that can be inscribed in the circle.

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Solution:

For the largest square inscribed in a circle, the diagonal of the square is equal to the diameter of the circle.

Let the side of the inscribed square be $s$.

The diameter of the circle is $d = 2r$.

The diagonal of a square with side $s$ is given by the Pythagorean theorem or simply $s\sqrt{2}$.

Since the diagonal of the inscribed square is equal to the diameter of the circle:

Now, we solve for the side $s$ from equation (i):

To rationalize the denominator:

$$s = \frac{2r\sqrt{2}}{2}$$

$$s = r\sqrt{2}$$

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The area of the square with side $s$ is given by $A_{square} = s^2$.

Substitute the value of $s$ we found:

$$A_{square} = r^2 \times (\sqrt{2})^2$$

$$A_{square} = r^2 \times 2$$

$$A_{square} = 2r^2$$

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The area of the largest square that can be inscribed in a circle of radius $r$ is $2r^2$.

Comparing this result with the given options, we find that it matches option (B).

Therefore, the correct option is (B) $2r^2$.

Given:

Radius of the sector $= r$.

Area of the sector $= A$.

Length of the corresponding arc $= l$.

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To Find:

The relationship between $A$, $r$, and $l$.

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Solution:

Let the central angle of the sector be $\theta$ in radians.

The formula for the area of a sector with radius $r$ and central angle $\theta$ (in radians) is:

The formula for the length of an arc ($l$) of a sector with radius $r$ and central angle $\theta$ (in radians) is:

From equation (ii), we can express $\theta$ in terms of $l$ and $r$ (assuming $r \neq 0$):

Now, substitute this expression for $\theta$ into the area formula (equation (i)):

Simplify the expression:

$$A = \frac{1}{2} r l$$

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The relationship between the area of a sector ($A$), the radius ($r$), and the arc length ($l$) is $A = \frac{1}{2} r l$.

Comparing this result with the given options, we find that it matches option (A).

Therefore, the correct option is (A) $\frac{1}{2} r l$.

Let's analyze each given statement:

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Statement (A): Area of major segment + Area of minor segment = Area of circle

A chord divides a circle into two segments: a minor segment and a major segment. The union of these two segments forms the entire circle. Therefore, the sum of their areas is equal to the area of the circle.

Area of Circle = Area of Minor Segment + Area of Major Segment.

This statement is True.

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Statement (B): Area of sector - Area of triangle = Area of segment

A segment is the region bounded by a chord and the arc it subtends. The corresponding sector is bounded by the two radii connecting the center to the endpoints of the arc and the arc itself. The triangle formed by these two radii and the chord is part of the sector. The area of the minor segment is obtained by subtracting the area of this triangle from the area of the corresponding minor sector.

Area of Minor Segment = Area of Minor Sector - Area of Triangle (formed by radii and chord).

This statement is True (specifically for the minor segment and minor sector/triangle formed by the same chord and radii).

---

Statement (C): Area of minor sector + Area of major sector = Area of circle

Two radii divide a circle into two sectors: a minor sector and a major sector. The union of these two sectors forms the entire circle. Therefore, the sum of their areas is equal to the area of the circle.

Area of Circle = Area of Minor Sector + Area of Major Sector.

This statement is True.

---

Since statements (A), (B), and (C) are all true statements regarding areas within a circle, option (D) which states "All of the above" is the correct choice.

---

Therefore, the correct option is (D) All of the above.

Given:

Radius of the sector $= r$.

Central angle of the sector $= \theta$ (in degrees).

---

To Find:

The perimeter of the sector.

---

Solution:

The perimeter of a sector is the total length of its boundary.

A sector is bounded by two radii and the arc connecting their endpoints.

The length of the two radii is $r + r = 2r$.

The length of the arc of the sector with radius $r$ and central angle $\theta$ (in degrees) is given by the formula:

The perimeter of the sector is the sum of the lengths of the two radii and the length of the arc:

Perimeter $= 2r + \frac{\theta}{360^\circ} \times 2\pi r$

---

Comparing this formula with the given options, we find that it matches option (B).

Option (A) is only the arc length.

Options (C) and (D) involve the area formula ($\pi r^2$) which is incorrect for perimeter.

---

Therefore, the correct option is (B) $2r + \frac{\theta}{360} \times 2\pi r$.

Given:

Radius of the circular table cover, $r = 28 \text{ cm}$.

Number of equal designs $= 6$.

Central angle subtended by the chord of each design (segment) $= 60^\circ$.

Cost of making designs $= \textsf{₹} 0.35$ per $\text{ cm}^2$.

---

To Find:

The total cost of making the six designs.

---

Solution:

The designs are segments of the circle. The area of each segment is the area of the corresponding sector minus the area of the triangle formed by the two radii and the chord.

The central angle for each of the 6 equal designs is $\frac{360^\circ}{6} = 60^\circ$. This confirms the given angle.

The triangle formed by the two radii (each of length $r$) and the chord subtending a $60^\circ$ angle at the center is an equilateral triangle with side length $r$.

---

First, calculate the area of one sector with radius $r = 28 \text{ cm}$ and central angle $\theta = 60^\circ$. We will use $\pi = \frac{22}{7}$.

Simplify the fraction $\frac{60}{360} = \frac{1}{6}$.

$A_{sector} = \frac{1}{6} \times \frac{22}{7} \times (28 \times 28) \text{ cm}^2$

$$A_{sector} = \frac{11 \times 112}{3} = \frac{1232}{3} \text{ cm}^2$$

---

Next, calculate the area of the equilateral triangle with side length $r = 28 \text{ cm}$. The problem provides the formula $A_{triangle} = \frac{\sqrt{3}}{4}r^2$. Based on the options, it seems $\sqrt{3}$ is approximated as $1.7$.

$$A_{triangle} = \frac{1.7}{4} \times 784 \text{ cm}^2$$

$$A_{triangle} = 1.7 \times \frac{784}{4} \text{ cm}^2 = 1.7 \times 196 \text{ cm}^2$$

$$A_{triangle} = 333.2 \text{ cm}^2$$

---

Now, calculate the area of one segment (one design).

Evaluate the area of the segment from equation (ii):

$\frac{1232}{3} \approx 410.6666...$

$$A_{segment} \approx 410.667 - 333.2 = 77.467 \text{ cm}^2$$

---

There are 6 such equal designs. The total area of the designs is:

Total Area $\approx 464.802 \text{ cm}^2$

---

Finally, calculate the cost of making the designs at the rate of $\textsf{₹} 0.35$ per $\text{ cm}^2$.

Cost $\approx 162.6807$

Rounding to two decimal places, the cost is approximately $\textsf{₹} 162.68$.

---

Comparing this result with the given options, we find that it matches option (A).

Therefore, the correct option is (A) $\textsf{₹} 162.68$ (approx).

Given:

Radius of the first circle $= r_1$.

Radius of the second circle $= r_2$.

---

To Find:

The radius ($R$) of a new circle whose circumference is equal to the sum of the circumferences of the two given circles.

---

Solution:

The circumference of the first circle is $C_1 = 2\pi r_1$.

The circumference of the second circle is $C_2 = 2\pi r_2$.

Let the radius of the new circle be $R$. Its circumference is $C_{new} = 2\pi R$.

According to the problem, the circumference of the new circle is the sum of the circumferences of the two given circles:

Substitute the formulas for the circumferences into equation (i):

Factor out $2\pi$ from the right side:

$2\pi R = 2\pi (r_1 + r_2)$

Divide both sides by $2\pi$ (since $2\pi \neq 0$):

$$R = r_1 + r_2$$

---

The radius of the new circle is $r_1 + r_2$.

Comparing this result with the given options, we find that it matches option (A).

Therefore, the correct option is (A) $r_1 + r_2$.

Given:

Radius of the circle formed by bending the wire, $r = 28 \text{ cm}$.

The wire is rebent into the shape of a square.

Value of $\pi = \frac{22}{7}$.

---

To Find:

The side length of the square.

---

Solution:

When a wire is bent into different shapes, the total length of the wire remains constant. The length of the wire is equal to the perimeter of the shape it forms.

Length of wire = Circumference of the circle

Length of wire = Perimeter of the square

---

Calculate the circumference of the circle with radius $r = 28 \text{ cm}$ using $\pi = \frac{22}{7}$.

Simplify the expression from equation (i):

Cancel out the common factor 7:

Circumference $= 2 \times 22 \times 4 \text{ cm}$

Circumference $= 44 \times 4 \text{ cm}$

Circumference $= 176 \text{ cm}$

The length of the wire is $176 \text{ cm}$.

---

Now, the wire is rebent into a square. Let the side length of the square be $s$.

The perimeter of the square is $4s$.

Since the length of the wire is equal to the perimeter of the square:

To find $s$, divide both sides by 4:

Perform the division from equation (ii):

$s = 44 \text{ cm}$

---

The side length of the square is $44 \text{ cm}$.

Comparing this result with the given options, we find that it matches option (B).

Therefore, the correct option is (B) 44 cm.

Given:

Side length of the square $= a$.

A circle is drawn inside the square.

---

To Find:

The area of the largest circle that can be drawn inside the square.

---

Solution:

For the largest circle to be inscribed in a square, the circle must be tangent to all four sides of the square.

In this case, the diameter of the circle is equal to the side length of the square.

Let the radius of the inscribed circle be $r$.

The diameter of the circle is $d = 2r$.

Given that the diameter is equal to the side of the square:

So, $2r = a$.

The radius of the largest inscribed circle is:

The area of a circle with radius $r$ is given by the formula $A_{circle} = \pi r^2$.

Substitute the value of $r$ from equation (i) into the area formula:

$$A_{circle} = \pi \left(\frac{a^2}{2^2}\right)$$

$$A_{circle} = \pi \frac{a^2}{4}$$

$$A_{circle} = \frac{\pi a^2}{4}$$

---

The area of the largest circle that can be drawn inside a square of side $a$ is $\frac{\pi a^2}{4}$.

Comparing this result with the given options, we find that it matches option (B).

Therefore, the correct option is (B) $\pi a^2 / 4$.

Given:

Radius of the wheel, $r = 0.25 \text{ m}$.

Total distance to be traveled, $D = 11 \text{ km}$.

Value of $\pi = \frac{22}{7}$.

---

To Find:

The number of revolutions the wheel will make to travel the given distance.

---

Solution:

The distance covered by a wheel in one revolution is equal to its circumference.

The circumference of the wheel is given by the formula $C = 2\pi r$.

Substitute the given radius and the value of $\pi$ into the formula:

Convert the radius from decimal to fraction: $0.25 = \frac{1}{4}$.

Simplify the expression from equation (i):

$$C = \frac{22}{14} \text{ m} = \frac{11}{7} \text{ m}$$

The circumference of the wheel is $\frac{11}{7} \text{ m}$.

---

The total distance is given in kilometers. We need to convert it to meters to match the unit of circumference.

$1 \text{ km} = 1000 \text{ m}$.

$D = 11000 \text{ m}$.

---

The number of revolutions is the total distance traveled divided by the distance covered in one revolution (circumference).

Substitute the values of $D$ and $C$ into equation (ii):

To divide by a fraction, multiply by its reciprocal:

Cancel out the common factor 11:

Number of revolutions $= 1000 \times 7$

Number of revolutions $= 7000$

---

The number of revolutions the wheel will make to travel a distance of 11 km is 7000.

Comparing this result with the given options, we find that it matches option (A).

Therefore, the correct option is (A) 7000.

Given:

Circumference of the circle, $C = 22 \text{ cm}$.

Value of $\pi = \frac{22}{7}$.

---

To Find:

The area of a quadrant of the circle.

---

Solution:

Let the radius of the circle be $r$. The formula for the circumference of a circle is $C = 2\pi r$.

Substitute the given circumference and the value of $\pi$ into the formula:

Simplify the right side of equation (i):

To find $r$, multiply both sides by $\frac{7}{44}$:

Cancel out the common factor 22:

$$r = \frac{7}{2} \text{ cm} = 3.5 \text{ cm}$$

The radius of the circle is $3.5 \text{ cm}$.

---

A quadrant of a circle is a sector with a central angle of $90^\circ$, which is one-fourth of the total area of the circle.

The formula for the area of a circle is $A_{circle} = \pi r^2$.

The area of a quadrant is $A_{quadrant} = \frac{1}{4} \times A_{circle} = \frac{1}{4} \pi r^2$.

Substitute the value of $r$ and $\pi$ into the formula:

Calculate $\left(\frac{7}{2}\right)^2 = \frac{49}{4}$.

Simplify the expression:

$$A_{quadrant} = \frac{11 \times 7}{2 \times 4} \text{ cm}^2 = \frac{77}{8} \text{ cm}^2$$

Calculate the decimal value:

$$\frac{77}{8} = 9.625 \text{ cm}^2$$

---

The area of the quadrant of a circle with circumference 22 cm is $9.625 \text{ cm}^2$. This value is not among the given options.

---

Let's examine the provided options:

• Option (A) $38.5 \text{ cm}^2$: This is the area of the full circle with radius $3.5 \text{ cm}$ ($A = \pi (3.5)^2 = 38.5$).
• Option (B) $19.25 \text{ cm}^2$: This is the area of a semicircle with radius $3.5 \text{ cm}$ ($\frac{1}{2} \times 38.5 = 19.25$).
• Option (C) $77 \text{ cm}^2$: This is the area of a circle with radius $\sqrt{77 \times 7 / 22} = \sqrt{49/2} = 7/\sqrt{2} \approx 4.95 \text{ cm}$. Or it is the area of a semicircle with radius $7 \text{ cm}$.
• Option (D) $154 \text{ cm}^2$: This is the area of a circle with radius $7 \text{ cm}$.

Based on the calculation from the given circumference, the correct area of the quadrant is $9.625 \text{ cm}^2$. Since this is not an option, there appears to be an error in the question or the provided options.

However, option (A) $38.5 \text{ cm}^2$ is the area of the full circle with the given circumference.

If the question intended the circumference to be $44 \text{ cm}$ (so $r=7 \text{ cm}$), then the area of the quadrant would be $\frac{1}{4} \pi (7)^2 = \frac{1}{4} \times \frac{22}{7} \times 49 = \frac{11 \times 7}{2} = \frac{77}{2} = 38.5 \text{ cm}^2$, which is option (A).

Given the likely context of geometry problems in this format, it is probable that the intended circumference was $44 \text{ cm}$ or the question intended to ask for the area of the circle.

Assuming there is a typo and the question implicitly expects one of the provided options to be correct, and considering option (A) is the area of the full circle corresponding to the given circumference, we highlight this relationship. However, the question explicitly asks for the area of a quadrant.

Since a correct option is not available based on the strict interpretation of the problem, we state the calculated answer and the relation of option (A) to the calculated radius.

---

Calculated Area of Quadrant $= 9.625 \text{ cm}^2$.

Option (A) $38.5 \text{ cm}^2$ is the Area of the Circle with the given circumference.

If we are forced to choose from the given options despite the discrepancy, option (A) is the area of the circle, not the quadrant.

Given:

Area of sector $= \frac{1}{6} \times$ Area of circle.

---

To Find:

The central angle of the sector.

---

Solution:

Let the radius of the circle be $r$ and the central angle of the sector be $\theta$ (in degrees).

The area of the circle is $A_{circle} = \pi r^2$.

The area of the sector with angle $\theta$ is $A_{sector} = \frac{\theta}{360^\circ} \times \pi r^2$.

According to the given information:

Substitute the formulas for the areas:

Assuming $r > 0$, we can divide both sides of equation (i) by $\pi r^2$:

Now, solve for $\theta$ by multiplying both sides by $360^\circ$:

Calculate the value from equation (ii):

$\theta = 60^\circ$

---

The angle of the sector is $60^\circ$.

Comparing this result with the given options, we find that it matches option (C).

Therefore, the correct option is (C) $60^\circ$.

Given:

Radius of the circular park (inner circle), $r_{inner} = 14 \text{ metres}$.

Width of the path $= 7 \text{ metres}$.

The path surrounds the park.

Value of $\pi = \frac{22}{7}$ (implied by option C calculation).

---

To Find:

The area of the path.

---

Solution:

The path surrounding the circular park forms a circular ring, also known as an annulus.

The outer boundary of the path is a circle with a larger radius, and the inner boundary is the circle of the park.

The radius of the outer circle, $r_{outer}$, is the sum of the radius of the park and the width of the path.

From equation (i):

---

The area of the path is the difference between the area of the outer circle and the area of the inner circle (the park).

The area of a circle is given by the formula $\pi r^2$.

Substitute the values of $r_{outer}$ and $r_{inner}$:

Calculate the squares:

$(21)^2 = 441$

$(14)^2 = 196$

So, $A_{path} = \pi (441) - \pi (196)$

Factor out $\pi$:

Perform the subtraction $441 - 196$:

$$\begin{array}{cccc} & \xcancel{4}^3 & \xcancel{4}^{13} & 11 \\ - & 1 & 9 & 6 \\ \hline & 2 & 4 & 5 \\ \hline \end{array}$$

So, $A_{path} = 245\pi \text{ m}^2$.

---

Now, substitute the value of $\pi = \frac{22}{7}$ to get the numerical area.

Simplify the expression:

$$A_{path} = 35 \times 22 \text{ m}^2$$

Perform the multiplication $35 \times 22$:

$$\begin{array}{cc}& & 3 & 5 \\ \times & & 2 & 2 \\ \hline && 7 & 0 \\ & 7 & 0 & \times \\ \hline & 7 & 7 & 0 \\ \hline \end{array}$$

---

The area of the path is $770 \text{ m}^2$.

Reviewing the options:

• (A) Area of outer circle - Area of inner circle: This is the correct formula/concept.
• (B) $\pi (14+7)^2 - \pi (14)^2 = \pi (21^2 - 14^2) = \pi (441 - 196) = 245\pi$: This shows the correct substitution and algebraic simplification.
• (C) $245 \times \frac{22}{7} = 35 \times 22 = 770 \text{ m}^2$: This shows the correct substitution of $\pi$ and the final numerical calculation.

All options (A), (B), and (C) present correct steps or calculations involved in finding the area of the path.

Therefore, the statement "All of the above are correct steps or calculation" is true.

---

The correct option is (D) All of the above are correct steps or calculation.

Given:

A semicircle with radius $r$.

---

To Find:

The formula for the perimeter of the semicircle.

---

Solution:

The perimeter of a shape is the total length of its boundary.

A semicircle is formed by cutting a circle into two equal halves along its diameter.

The boundary of a semicircle consists of two parts:

1. The curved part, which is half the circumference of the full circle.
2. The straight part, which is the diameter of the circle.

The circumference of a full circle with radius $r$ is $2\pi r$.

The length of the curved part of the semicircle is half of the circumference:

The diameter of the circle with radius $r$ is $d = 2r$.

The length of the straight part (diameter) of the semicircle is $2r$.

---

The perimeter of the semicircle is the sum of the lengths of the curved part and the straight part:

The formula can also be written by factoring out $r$:

Perimeter of semicircle $= r(\pi + 2)$

---

Comparing the formula derived in equation (i) with the given options:

Option (A) $\pi r$ is only the length of the curved part.

Option (B) $\pi r + 2r$ matches the derived formula for the perimeter.

Option (C) $\frac{1}{2} \pi r^2$ is the formula for the area of a semicircle.

Option (D) $2\pi r$ is the formula for the circumference of a full circle.

---

Therefore, the correct option is (B) $\pi r + 2r$.

Given:

Radius of the first circle $= r_1$.

Radius of the second circle $= r_2$.

The area of a new circle is equal to the sum of the areas of the two given circles.

---

To Find:

The radius of the new circle, let's call it $R$.

---

Solution:

The area of the first circle is $A_1 = \pi r_1^2$.

The area of the second circle is $A_2 = \pi r_2^2$.

The area of the new circle with radius $R$ is $A_{new} = \pi R^2$.

According to the problem statement, the area of the new circle is the sum of the areas of the two circles:

Substitute the formulas for the areas into equation (i):

Factor out $\pi$ from the terms on the right side:

Since $\pi$ is a non-zero constant, we can divide both sides of the equation by $\pi$:

To find the radius $R$, we take the square root of both sides of equation (ii). Since $R$ represents a radius, it must be a non-negative value.

---

The radius of the circle whose area is equal to the sum of the areas of the two circles is $\sqrt{r_1^2 + r_2^2}$.

Comparing this result with the given options, we find that it matches option (C).

Therefore, the correct option is (C) $\sqrt{r_1^2 + r_2^2}$.

Given:

Central angle of the sector $= 120^\circ$.

---

To Find:

The fraction of the circle that the sector represents.

---

Solution:

The fraction of a circle that a sector represents is given by the ratio of its central angle to the total angle in a circle ($360^\circ$).

Substitute the given central angle and the total angle:

Simplify the fraction from equation (i):

$$ \text{Fraction} = \frac{1}{3} $$

So, a sector with an angle of $120^\circ$ is one-third of the circle.

---

Comparing this result with the given options, we find that it matches option (A).

Therefore, the correct option is (A) One-third.

Given:

Total cost of fencing = $\textsf{₹} 5280$.

Rate of fencing = $\textsf{₹} 24$ per metre.

The field is circular.

Value of $\pi = \frac{22}{7}$.

---

To Find:

The radius of the circular field, $r$.

---

Solution:

The total length of the fence required is the circumference of the circular field.

The total cost of fencing is given by the formula:

Let $C$ be the circumference of the field. Substitute the given values into equation (i):

To find the Circumference $C$, divide the total cost by the rate per metre:

Perform the division:

$$C = 220 \text{ metres}$$

---

The circumference of a circle with radius $r$ is given by the formula $C = 2\pi r$.

Substitute the calculated circumference ($C=220 \text{ m}$) and the given value of $\pi = \frac{22}{7}$ into this formula:

Simplify the equation:

To find the radius $r$, multiply both sides of the equation by the reciprocal of $\frac{44}{7}$, which is $\frac{7}{44}$:

Simplify the expression from equation (ii):

$$r = 5 \times 7 \text{ metres}$$

$$r = 35 \text{ metres}$$

---

The radius of the circular field is 35 m.

Comparing this result with the given options, we find that it matches option (A).

Therefore, the correct option is (A) 35 m.

Let's evaluate each statement:

---

Statement (A): A chord is a part of a segment.

A segment of a circle is the region enclosed by a chord and the arc it subtends. The chord is a boundary of the segment, not a part of the two-dimensional area of the segment itself.

This statement is False.

---

Statement (B): A radius is a part of a diameter.

A diameter of a circle is a line segment that passes through the center and has its endpoints on the circle. Its length is $2r$, where $r$ is the radius. A radius is a line segment from the center to a point on the circle. A diameter consists of two radii joined at the center.

So, a radius is indeed a part of the line segment that forms the diameter.

This statement is True.

---

Statement (C): An arc is a part of a chord.

An arc is a curved section of the circumference of a circle. A chord is a straight line segment connecting two points on the circle. An arc and a chord subtend the same two points on the circle, but they are different geometric entities (a curve vs. a straight line). An arc is not a part of a chord.

This statement is False.

---

Statement (D): A sector is a part of a segment.

A sector is the region bounded by two radii and the arc between them. A segment is the region bounded by a chord and the arc between them. These are different regions of the circle. For example, the area of a minor segment is found by subtracting the area of the triangle (formed by the two radii and the chord) from the area of the corresponding sector.

This statement is False.

---

Based on the analysis, only statement (B) is true.

Therefore, the correct option is (B) A radius is a part of a diameter.

Given:

Dimensions of the rectangular field: 20m $\times$ 14m.

Length of the rope tying the cow = 7m.

The cow is tied at one corner of the field.

---

To Find:

The area the cow can graze.

---

Solution:

The cow is tied at a corner of the rectangular field with a rope of length 7m.

The rope acts as the radius, limiting the area the cow can reach to a circle of radius 7m centered at the corner where it is tied.

However, the cow is within a rectangular field, and the corner of a rectangle forms a right angle, which is $90^\circ$.

The area the cow can graze is the part of the circle of radius 7m that lies within the boundaries of the field at the corner.

Since the corner of the rectangular field is $90^\circ$, the grazing area is restricted to the sector of the circle defined by this angle.

This area is a sector of a circle with radius equal to the length of the rope ($r = 7\text{ m}$) and the central angle equal to the angle of the corner of the field ($\theta = 90^\circ$).

Such a sector, with a $90^\circ$ central angle, is also known as a quadrant of a circle.

---

Let's compare this conclusion with the given options:

(A) Area of a circle with radius 7m: This would be the case if the cow was in an open field without boundaries.

(B) Area of a sector with radius 7m and angle $90^\circ$: This accurately describes the grazing area constrained by the rope length and the $90^\circ$ corner of the rectangular field.

(C) Area of a square with side 7m: The grazing area is curved, not square.

(D) Area of a semicircle with radius 7m: A semicircle corresponds to a $180^\circ$ angle, not the $90^\circ$ corner of a rectangle.

---

Therefore, the area the cow can graze is the area of a sector with radius 7m and angle $90^\circ$.

The correct option is (B) Area of a sector with radius 7m and angle $90^\circ$.

Given:

The circumference of a circle exceeds its diameter by $16.8 \text{ cm}$.

The value of $\pi$ to be used is $\frac{22}{7}$.

---

To Find:

The radius of the circle.

---

Solution:

Let the radius of the circle be $r \text{ cm}$.

The circumference of the circle, $C$, is given by the formula $C = 2\pi r$.

The diameter of the circle, $D$, is given by the formula $D = 2r$.

According to the problem statement, the circumference exceeds the diameter by $16.8 \text{ cm}$. This can be written as:

Substitute the formulas for $C$ and $D$ into equation (i):

Now, we need to solve this equation for $r$. First, bring all terms involving $r$ to one side:

Factor out $2r$ from the left side:

Divide both sides by 2:

Substitute the given value of $\pi = \frac{22}{7}$ into the equation:

Simplify the expression inside the parenthesis:

To find $r$, multiply both sides by the reciprocal of $\frac{15}{7}$, which is $\frac{7}{15}$:

Convert $8.4$ into a fraction to simplify the calculation:

Simplify the fraction by cancelling common factors. Divide 84 and 10 by 2, and 42 and 15 by 3:

Convert the fraction to a decimal:

---

Final Answer:

The radius of the circle is $3.92 \text{ cm}$.

Given:

Diameter of the bicycle wheel, $D = 84 \text{ cm}$.

Total distance to be covered, $S = 132 \text{ m}$.

Value of $\pi = \frac{22}{7}$.

---

To Find:

The number of complete revolutions the wheel must make.

---

Solution:

First, calculate the radius of the wheel.

Radius, $r = \frac{\text{Diameter}}{2}$

The distance covered by the wheel in one complete revolution is equal to its circumference.

The circumference of the wheel, $C$, is given by the formula $C = 2\pi r$.

Using the given value of $\pi = \frac{22}{7}$:

Simplify the expression:

The circumference of the wheel is $264 \text{ cm}$. This is the distance covered in one revolution.

The total distance to be covered is $132 \text{ m}$. Convert this distance to centimeters (since the circumference is in cm).

$1 \text{ m} = 100 \text{ cm}$

Let $n$ be the number of complete revolutions required to cover the distance of $13200 \text{ cm}$.

Total distance = Number of revolutions $\times$ Circumference of the wheel

To find $n$, divide the total distance by the circumference:

We can simplify this fraction. Both numbers are divisible by 12:

Now, divide 1100 by 22:

---

Final Answer:

The wheel must make 50 complete revolutions to cover a distance of $132 \text{ m}$.

Given:

Radius of the circle, $r = 6 \text{ cm}$.

Angle of the sector, $\theta = 60^\circ$.

Value of $\pi = \frac{22}{7}$.

---

To Find:

The area of the sector.

---

Solution:

The area of a sector of a circle with radius $r$ and angle $\theta$ (in degrees) is given by the formula:

Area of sector $= \frac{\theta}{360^\circ} \times \pi r^2$

Substitute the given values into the formula:

Simplify the fraction $\frac{60^\circ}{360^\circ}$:

Now, perform the multiplication. We can cancel out the 6 in the denominator with the 36:

The area can also be expressed as a decimal:

However, leaving the answer as an improper fraction $\frac{132}{7}$ is often preferred when $\pi = \frac{22}{7}$ is used.

---

Final Answer:

The area of the sector is $\frac{132}{7} \text{ cm}^2$.

Given:

Radius of the circle, $r = 14 \text{ cm}$.

The chord subtends a right angle at the center. This is the angle of the corresponding minor sector.

Angle of the minor sector, $\theta_{\text{minor}} = 90^\circ$.

Value of $\pi = \frac{22}{7}$.

---

To Find:

The area of the corresponding major sector.

---

Solution:

The angle of the major sector is the remaining angle of the circle after subtracting the angle of the minor sector from $360^\circ$.

The area of a sector of a circle with radius $r$ and angle $\theta$ (in degrees) is given by the formula:

Area of sector $= \frac{\theta}{360^\circ} \times \pi r^2$

Substitute the values for the major sector:

Simplify the fraction $\frac{270^\circ}{360^\circ}$:

Now, perform the multiplication. We can cancel out common factors:

Let's re-calculate step-by-step:

---

Final Answer:

The area of the corresponding major sector is $462 \text{ cm}^2$.

Given:

Radius of the circle, $r = 4 \text{ cm}$.

Angle of the minor sector, $\theta_{\text{minor}} = 30^\circ$.

Value of $\pi = 3.14$.

---

To Find:

The area of the minor sector.

The area of the corresponding major sector.

---

Solution:

The area of a sector of a circle with radius $r$ and angle $\theta$ (in degrees) is given by the formula:

Area of sector $= \frac{\theta}{360^\circ} \times \pi r^2$

First, calculate the area of the minor sector with angle $\theta_{\text{minor}} = 30^\circ$ and radius $r = 4 \text{ cm}$.

Simplify the fraction $\frac{30^\circ}{360^\circ}$:

Substitute this and the value of $r^2$ into equation (i):

Multiply the terms:

Cancel the common factor 4 from 16 and 12:

Now, calculate the angle of the corresponding major sector. The total angle in a circle is $360^\circ$.

Now, calculate the area of the major sector using the same formula with $\theta_{\text{major}} = 330^\circ$ and $r = 4 \text{ cm}$.

Simplify the fraction $\frac{330^\circ}{360^\circ}$:

Substitute this and the value of $r^2$:

Multiply the terms and cancel common factors:

Expressing the areas as decimals (rounded to two decimal places):

---

Final Answer:

The area of the minor sector is approximately $4.19 \text{ cm}^2$.

The area of the corresponding major sector is approximately $46.05 \text{ cm}^2$.

Given:

Length of the minute hand = Radius of the circle, $r = 14 \text{ cm}$.

Time duration = $5$ minutes.

---

To Find:

The area swept by the minute hand in $5$ minutes.

---

Solution:

The minute hand of a clock completes one full revolution ($360^\circ$) in $60$ minutes.

Angle swept by the minute hand in $60$ minutes $= 360^\circ$.

Angle swept by the minute hand in $1$ minute $= \frac{360^\circ}{60^\circ}$ per minute $= 6^\circ$ per minute.

Angle swept by the minute hand in $5$ minutes:

The area swept by the minute hand in $5$ minutes is the area of a sector of the circle with radius $r = 14 \text{ cm}$ and angle $\theta = 30^\circ$.

The formula for the area of a sector is:

Area of sector $= \frac{\theta}{360^\circ} \times \pi r^2$

Substitute the values of $\theta$, $r$, and $\pi = \frac{22}{7}$:

Simplify the fraction $\frac{30^\circ}{360^\circ}$:

Perform the calculation. Cancel common factors:

---

Final Answer:

The area swept by the minute hand in $5$ minutes is $\frac{154}{3} \text{ cm}^2$.

Given:

Side of the square, $s = 10 \text{ cm}$.

The square is inscribed in a circle.

Value of $\pi = 3.14$.

---

To Find:

The area of the circle.

---

Solution:

When a square is inscribed in a circle, the diagonal of the square is equal to the diameter of the circle.

Let the side of the square be $s$ and the diagonal be $d$. By the Pythagorean theorem, the diagonal of a square is given by:

Given the side of the square $s = 10 \text{ cm}$:

The diameter of the circle is equal to the diagonal of the inscribed square.

The radius of the circle, $r$, is half of the diameter:

The area of the circle is given by the formula $A = \pi r^2$.

Substitute the value of $r$ and $\pi = 3.14$:

Perform the multiplication:

---

Final Answer:

The area of the circle is $157 \text{ cm}^2$.

Given:

Radius of the sector (and circle), $r = 5.2 \text{ cm}$.

Perimeter of the sector $= 16.4 \text{ cm}$.

---

To Find:

The area of the sector.

---

Solution:

The perimeter of a sector of a circle consists of two radii and the arc length. Let the arc length of the sector be $L$.

Perimeter of sector $=$ Radius $+$ Radius $+$ Arc length

Substitute the given value of the radius, $r = 5.2 \text{ cm}$:

Solve for the arc length $L$:

The area of a sector can be calculated using the formula that relates the arc length and the radius:

Area of sector $= \frac{1}{2} \times \text{Arc length} \times \text{Radius}$

Substitute the calculated arc length $L = 6 \text{ cm}$ and the given radius $r = 5.2 \text{ cm}$:

Perform the multiplication:

---

Final Answer:

The area of the sector is $15.6 \text{ cm}^2$.

Given:

Radius of the first circle, $r_1 = 19 \text{ cm}$.

Radius of the second circle, $r_2 = 9 \text{ cm}$.

---

To Find:

The radius of the circle which has circumference equal to the sum of the circumferences of the two circles.

---

Solution:

Let $C_1$ be the circumference of the first circle, $C_2$ be the circumference of the second circle, and $C$ be the circumference of the new circle with radius $R$.

The formula for the circumference of a circle with radius $r$ is $C = 2\pi r$.

The circumference of the first circle is $C_1 = 2\pi r_1$.

The circumference of the second circle is $C_2 = 2\pi r_2$.

The circumference of the new circle is $C = 2\pi R$.

According to the problem, the circumference of the new circle is equal to the sum of the circumferences of the two given circles:

Substitute the circumference formulas into equation (i):

Factor out $2\pi$ from the right side of the equation:

Divide both sides of the equation by $2\pi$:

Now, substitute the given values for $r_1$ and $r_2$:

---

Final Answer:

The radius of the circle which has circumference equal to the sum of the circumferences of the two circles is $28 \text{ cm}$.

Given:

Radius of the first circle, $r_1 = 8 \text{ cm}$.

Radius of the second circle, $r_2 = 6 \text{ cm}$.

A new circle has an area equal to the sum of the areas of the two given circles.

---

To Find:

The radius of the new circle.

---

Solution:

Let $A_1$ be the area of the first circle, $A_2$ be the area of the second circle, and $A$ be the area of the new circle with radius $R$.

The formula for the area of a circle with radius $r$ is $A = \pi r^2$.

The area of the first circle is $A_1 = \pi r_1^2$.

Substitute the value of $r_1$:

The area of the second circle is $A_2 = \pi r_2^2$.

Substitute the value of $r_2$:

The area of the new circle with radius $R$ is $A = \pi R^2$.

According to the problem, the area of the new circle is equal to the sum of the areas of the two given circles:

Substitute the area formulas and the calculated areas into equation (i):

Combine the terms on the right side:

Divide both sides of the equation by $\pi$:

Take the square root of both sides to find $R$. Since radius must be positive:

---

Final Answer:

The radius of the circle having area equal to the sum of the areas of the two circles is $10 \text{ cm}$.

Given:

Length of the rope, which is the radius of the grazing area, $r = 10 \text{ m}$.

Side of the square field $= 15 \text{ m}$.

The cow is tied at a corner of the square field.

Value of $\pi = 3.14$.

---

To Find:

The area of the field in which the cow can graze.

---

Solution:

Since the cow is tied at a corner of the square field, the area it can graze is a sector of a circle with the corner as the center and the length of the rope as the radius.

The angle of the sector is the angle of the corner of the square field, which is $90^\circ$.

Since the side of the square field ($15 \text{ m}$) is greater than the length of the rope ($10 \text{ m}$), the grazing area is limited only by the rope and the boundaries of the square field at that corner. The grazing area forms a quarter circle.

The area of a sector of a circle with radius $r$ and angle $\theta$ (in degrees) is given by the formula:

Area of sector $= \frac{\theta}{360^\circ} \times \pi r^2$

Substitute the given values into the formula:

Simplify the fraction $\frac{90^\circ}{360^\circ}$:

Perform the multiplication:

---

Final Answer:

The area of the field in which the cow can graze is $78.5 \text{ m}^2$.

Given:

Square ABCD with side $s = 14 \text{ cm}$.

Arcs are drawn with centers A, B, C, and D, such that each arc touches two other arcs.

Value of $\pi = \frac{22}{7}$.

---

To Find:

The area of the shaded region.

---

Solution:

Since the arcs are drawn with centers at the vertices of the square and each arc touches two other arcs, the radius of each arc is half the side length of the square.

The angle of the sector at each corner of the square is $90^\circ$ (since each angle of a square is $90^\circ$).

The area of the square is given by the formula Area $= \text{side}^2$.

The area of each sector is given by the formula:

Area of sector $= \frac{\theta}{360^\circ} \times \pi r^2$

Substitute the values for one sector:

Simplify the fraction and calculate the area:

There are four such sectors, one at each corner of the square. The sum of the areas of the four sectors is:

Alternatively, the four sectors, each with a $90^\circ$ angle, combine to form a full circle ($4 \times 90^\circ = 360^\circ$). The area of this full circle with radius $7 \text{ cm}$ is $\pi r^2 = \frac{22}{7} \times (7 \text{ cm})^2 = \frac{22}{7} \times 49 \text{ cm}^2 = 22 \times 7 \text{ cm}^2 = 154 \text{ cm}^2$. Both methods yield the same result.

The area of the shaded region is the area of the square minus the total area of the four sectors.

---

Final Answer:

The area of the shaded region is $42 \text{ cm}^2$.

Given:

Radius of the circle, $r = 10 \text{ cm}$.

Angle subtended by the chord at the center, $\theta = 90^\circ$.

Value of $\pi = 3.14$.

---

To Find:

The area of the minor segment.

---

Solution:

The area of the minor segment is the difference between the area of the corresponding sector and the area of the triangle formed by the radii and the chord.

The angle of the sector is given as $\theta = 90^\circ$.

The radius is $r = 10 \text{ cm}$.

The area of the sector is given by the formula:

Area of sector $= \frac{\theta}{360^\circ} \times \pi r^2$

Substitute the given values:

The triangle formed by the radii and the chord is a right-angled triangle since the angle at the center is $90^\circ$. The base and height of this triangle are equal to the radius $r$.

The area of the triangle is given by the formula:

Area of triangle $= \frac{1}{2} \times \text{base} \times \text{height}$

Substitute the value of $r$:

Now, calculate the area of the minor segment:

Area of minor segment $=$ Area of sector $-$ Area of triangle

---

Final Answer:

The area of the minor segment is $28.5 \text{ cm}^2$.

Given:

Circumference of the circle, $C = 88 \text{ cm}$.

Value of $\pi = \frac{22}{7}$.

---

To Find:

The area of the circle.

---

Solution:

Let the radius of the circle be $r \text{ cm}$.

The circumference of a circle is given by the formula $C = 2\pi r$.

We are given that the circumference is $88 \text{ cm}$. So, we can write:

Substitute the given value of $\pi = \frac{22}{7}$:

To find $r$, multiply both sides of the equation by $\frac{7}{44}$:

Simplify the expression by cancelling common factors (88 is divisible by 44):

Now that we have the radius, we can find the area of the circle using the formula $A = \pi r^2$.

Substitute the values of $\pi$ and $r$:

Simplify the expression by cancelling common factors (196 is divisible by 7):

Perform the multiplication:

---

Final Answer:

The area of the circle is $616 \text{ cm}^2$.

Given:

Area of the circle, $A = 154 \text{ cm}^2$.

Value of $\pi = \frac{22}{7}$.

---

To Find:

The circumference of the circle.

---

Solution:

Let the radius of the circle be $r \text{ cm}$.

The area of a circle is given by the formula $A = \pi r^2$.

We are given that the area is $154 \text{ cm}^2$. Substitute the values into the formula:

Substitute the given value of $\pi = \frac{22}{7}$ into equation (i):

To find $r^2$, multiply both sides by the reciprocal of $\frac{22}{7}$, which is $\frac{7}{22}$:

Simplify the expression by cancelling common factors (154 is divisible by 22):

Take the square root of both sides to find $r$. Since radius must be positive:

Now, we need to find the circumference of the circle. The formula for the circumference is $C = 2\pi r$.

Substitute the values of $\pi = \frac{22}{7}$ and $r = 7 \text{ cm}$:

Simplify the expression by cancelling common factors:

---

Final Answer:

The circumference of the circle is $44 \text{ cm}$.

Given:

Circumference of the circle, $C = 22 \text{ cm}$.

Value of $\pi = \frac{22}{7}$.

---

To Find:

The area of a quadrant of the circle.

---

Solution:

Let the radius of the circle be $r \text{ cm}$.

The circumference of a circle is given by the formula $C = 2\pi r$.

We are given that the circumference is $22 \text{ cm}$. Substitute the values into the formula:

Substitute the given value of $\pi = \frac{22}{7}$:

To find $r$, multiply both sides by the reciprocal of $\frac{44}{7}$, which is $\frac{7}{44}$:

Simplify the expression by cancelling common factors (22 is a factor of 44):

A quadrant of a circle is a sector covering one-fourth of the circle. Its angle at the center is $90^\circ$. The area of a quadrant is $\frac{1}{4}$ of the area of the full circle.

The area of the circle is given by the formula $A = \pi r^2$.

The area of the quadrant is $\frac{1}{4} \times \pi r^2$.

Substitute the values of $\pi = \frac{22}{7}$ and $r = \frac{7}{2} \text{ cm}$:

Simplify the expression by cancelling common factors:

Let's re-calculate step-by-step:

Simplify the fraction by dividing numerator and denominator by 2:

The area can also be expressed as a decimal:

---

Final Answer:

The area of a quadrant of the circle is $\frac{77}{8} \text{ cm}^2$ or $9.625 \text{ cm}^2$.

Given:

Radius of the circular park (inner circle), $r_1 = 28 \text{ m}$.

Width of the circular path $= 3.5 \text{ m}$.

The path runs around the park on the outside.

Value of $\pi = \frac{22}{7}$.

---

To Find:

The area of the circular path.

---

Solution:

The circular path forms a ring around the park. The area of the path is the difference between the area of the larger circle (park plus path) and the area of the smaller circle (the park itself).

The radius of the larger circle, $r_2$, is the radius of the park plus the width of the path.

The area of the inner circle (park) is $A_1 = \pi r_1^2$.

The area of the larger circle (park plus path) is $A_2 = \pi r_2^2$.

The area of the path is $A_{\text{path}} = A_2 - A_1$.

Factor out $\pi$:

Substitute the values of $r_1$, $r_2$, and $\pi = \frac{22}{7}$:

We can use the difference of squares formula: $a^2 - b^2 = (a-b)(a+b)$. Here $a = 31.5$ and $b = 28$.

Now, calculate the area of the path:

Convert decimals to fractions for easier calculation with $\pi = \frac{22}{7}$:

Simplify the expression by cancelling common factors:

Perform the multiplication $11 \times 119$:

Convert the fraction to a decimal:

---

Final Answer:

The area of the path is $654.5 \text{ m}^2$.

Given:

The difference between the circumference and radius of a circle is $37 \text{ cm}$.

Value of $\pi = \frac{22}{7}$.

---

To Find:

The area of the circle.

---

Solution:

Let the radius of the circle be $r \text{ cm}$ and the circumference be $C \text{ cm}$.

We are given that the difference between the circumference and radius is $37 \text{ cm}$.

The formula for the circumference of a circle with radius $r$ is $C = 2\pi r$.

Substitute this formula for $C$ into equation (i):

Factor out $r$ from the left side:

Substitute the given value of $\pi = \frac{22}{7}$:

Simplify the expression inside the parenthesis:

To find $r$, multiply both sides by the reciprocal of $\frac{37}{7}$, which is $\frac{7}{37}$:

Simplify the expression by cancelling common factors (37):

Now that we have the radius, we can find the area of the circle using the formula $A = \pi r^2$.

Substitute the values of $\pi = \frac{22}{7}$ and $r = 7 \text{ cm}$:

Simplify the expression by cancelling common factors (49 is divisible by 7):

---

Final Answer:

The area of the circle is $154 \text{ cm}^2$.

Given:

Radius of the circle, $r = 12 \text{ cm}$.

Angle subtended by the chord at the center, $\theta = 60^\circ$.

Value of $\pi = 3.14$.

Value of $\sqrt{3} = 1.73$.

---

To Find:

The area of the minor segment of the circle.

---

Solution:

The area of the minor segment is the difference between the area of the corresponding sector and the area of the triangle formed by the chord and the two radii to the endpoints of the chord.

The area of the sector with angle $\theta$ (in degrees) and radius $r$ is given by the formula:

Area of sector $= \frac{\theta}{360^\circ} \times \pi r^2$

Substitute the given values $r = 12 \text{ cm}$, $\theta = 60^\circ$, and $\pi = 3.14$:

Simplify the fraction $\frac{60^\circ}{360^\circ}$:

Perform the multiplication. We can cancel out the 6 with 144:

Calculate $3.14 \times 24$:

Now, consider the triangle formed by the two radii and the chord. The two radii are $r = 12 \text{ cm}$. The angle between them is $\theta = 60^\circ$. This is an isosceles triangle. Since the angle between the two equal sides is $60^\circ$, the triangle is equilateral. However, we can use the general formula for the area of a triangle with two sides and the included angle:

Area of triangle $= \frac{1}{2} r^2 \sin(\theta)$

Substitute the values $r = 12 \text{ cm}$ and $\theta = 60^\circ$:

We know that $\sin(60^\circ) = \frac{\sqrt{3}}{2}$. Substitute the values:

Use the given value $\sqrt{3} = 1.73$:

Calculate $36 \times 1.73$:

Now, calculate the area of the minor segment:

Area of minor segment $=$ Area of sector $-$ Area of triangle

Perform the subtraction:

---

Final Answer:

The area of the minor segment of the circle is $13.08 \text{ cm}^2$.

Given:

Radius of the circle, $r = 21 \text{ cm}$.

Angle subtended by the arc at the center, $\theta = 60^\circ$.

Value of $\pi = \frac{22}{7}$.

---

To Find:

1. The length of the arc.

2. The area of the sector formed by the arc.

---

Solution:

1. Length of the arc:

The length of an arc of a circle with radius $r$ and angle $\theta$ (in degrees) at the center is given by the formula:

Length of arc, $L = \frac{\theta}{360^\circ} \times 2\pi r$

Substitute the given values $r = 21 \text{ cm}$, $\theta = 60^\circ$, and $\pi = \frac{22}{7}$:

Simplify the fraction $\frac{60^\circ}{360^\circ}$:

Perform the multiplication and cancel common factors:

Let's calculate more cautiously step-by-step:

The length of the arc is $22 \text{ cm}$.

---

2. Area of the sector:

The area of a sector of a circle with radius $r$ and angle $\theta$ (in degrees) is given by the formula:

Area of sector, $A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2$

Substitute the given values $r = 21 \text{ cm}$, $\theta = 60^\circ$, and $\pi = \frac{22}{7}$:

Simplify the fraction $\frac{60^\circ}{360^\circ}$:

Perform the multiplication and cancel common factors:

The area of the sector is $231 \text{ cm}^2$.

---

Final Answer:

The length of the arc is $22 \text{ cm}$.

The area of the sector formed by the arc is $231 \text{ cm}^2$.

Given:

Perimeter of the semicircular protractor = $36 \text{ cm}$.

Value of $\pi = \frac{22}{7}$.

---

To Find:

The diameter of the semicircular protractor.

---

Solution:

Let the radius of the semicircular protractor be $r \text{ cm}$ and the diameter be $D \text{ cm}$.

The perimeter of a semicircular protractor consists of two parts: the semicircular arc and the diameter (straight edge).

The length of the semicircular arc is half the circumference of a full circle. Circumference of a full circle is $2\pi r$.

Length of semicircular arc $= \frac{1}{2} \times 2\pi r = \pi r$

The diameter $D = 2r$, so the radius $r = D/2$. The arc length can also be written in terms of diameter as $\pi (D/2)$.

The perimeter of the semicircular protractor is the sum of the length of the semicircular arc and the diameter.

Alternatively, using the diameter $D$:

We are given that the perimeter is $36 \text{ cm}$. So,

Substitute the given value of $\pi = \frac{22}{7}$ into equation (i):

Simplify the expression inside the parenthesis:

To find $D$, multiply both sides by the reciprocal of $\frac{18}{7}$, which is $\frac{7}{18}$:

Simplify the expression by cancelling common factors (36 is divisible by 18):

---

Final Answer:

The diameter of the semicircular protractor is $14 \text{ cm}$.

Given:

Radius of the circle, $r = 8 \text{ cm}$.

A square is inscribed in the circle.

---

To Find:

The area of the inscribed square.

---

Solution:

When a square is inscribed in a circle, the diagonal of the square is equal to the diameter of the circle.

The diameter of the circle, $D$, is twice the radius:

The diagonal of the inscribed square is equal to the diameter of the circle.

Let the side of the square be $s$. In a square, the relationship between the diagonal $d$ and the side $s$ is given by the Pythagorean theorem:

Substitute the value of the diagonal $d = 16 \text{ cm}$:

Solve for the side $s$:

To rationalize the denominator, multiply the numerator and denominator by $\sqrt{2}$:

The area of the square is given by the formula Area $= s^2$.

Substitute the value of $s$:

---

Final Answer:

The area of the square that can be inscribed in a circle of radius $8 \text{ cm}$ is $128 \text{ cm}^2$.

Given:

The plot is a rectangle ABCD with length AB = $20 \text{ m}$ and width BC = $14 \text{ m}$.

A semicircular grass lawn is attached to side BC.

Value of $\pi = \frac{22}{7}$.

---

To Find:

The area of the total plot.

---

Solution:

The total area of the plot is the sum of the area of the rectangle and the area of the semicircular grass lawn.

1. Area of the rectangle:

The dimensions of the rectangle are AB = $20 \text{ m}$ and BC = $14 \text{ m}$.

Area of rectangle = Length $\times$ Width

2. Area of the semicircular grass lawn:

The semicircle is attached to side BC. Therefore, BC is the diameter of the semicircle.

The radius of the semicircle, $r$, is half of the diameter.

The area of a semicircle is half the area of a full circle with the same radius.

Area of semicircle $= \frac{1}{2} \pi r^2$

Substitute the values $r = 7 \text{ m}$ and $\pi = \frac{22}{7}$:

Perform the calculation and cancel common factors:

3. Total area of the plot:

Total Area $=$ Area of rectangle $+$ Area of semicircle

---

Final Answer:

The area of the total plot is $357 \text{ m}^2$.

Given:

Square lawn ABCD with side $s = 56 \text{ m}$.

Two circular flower beds on two sides (assumed to be AD and BC), centered at O, the intersection of the diagonals.

Value of $\pi = \frac{22}{7}$.

---

To Find:

The sum of the areas of the lawn and the two flower beds.

---

Solution:

The total area required is the sum of the area of the square lawn and the areas of the two circular flower beds.

1. Area of the square lawn:

The side of the square lawn is $s = 56 \text{ m}$.

Area of square lawn $= \text{side}^2$

Calculate $56^2$:

2. Area of the two circular flower beds:

The center of each circular flower bed is O, the intersection point of the diagonals of the square. The flower beds are on two sides, say AD and BC.

The diagonals of a square bisect each other at right angles. The segments from the center O to the vertices A, B, C, D are equal in length. These lengths serve as the radius for the circular sectors forming the flower beds.

The diagonal of the square ABCD is $d = s\sqrt{2}$.

The radius of the circular sectors is half the diagonal.

The angle of the sector at the center O corresponding to side AD (or BC) is the angle between the diagonals, which is $90^\circ$ in a square.

The area of one circular flower bed (sector) is given by the formula:

Area of sector $= \frac{\theta}{360^\circ} \times \pi r^2$

Substitute the values $r = 28\sqrt{2} \text{ m}$, $\theta = 90^\circ$, and $\pi = \frac{22}{7}$:

Perform the calculation and cancel common factors:

Calculate $56 \times 22$:

There are two such flower beds.

3. Sum of the areas:

Total area $=$ Area of lawn $+$ Total area of flower beds

---

Final Answer:

The sum of the areas of the lawn and the flower beds is $5600 \text{ m}^2$.

Given:

Radius of the round table cover, $r = 28 \text{ cm}$.

Number of equal designs = 6.

Cost of making designs = $\textsf{₹} 0.35$ per $\text{cm}^2$.

Value of $\pi = \frac{22}{7}$.

Value of $\sqrt{3} = 1.7$.

---

To Find:

The total cost of making the six designs.

---

Solution:

The six equal designs are segments of the circle. Each design is formed by a chord and the corresponding arc. Since there are six equal designs, they divide the circle into six equal sectors. The angle subtended by each sector at the center is $\theta = \frac{360^\circ}{6} = 60^\circ$.

The area of each design (segment) is the difference between the area of the corresponding sector and the area of the triangle formed by the two radii and the chord.

1. Area of one sector:

The area of a sector with radius $r$ and angle $\theta$ (in degrees) is given by the formula:

Area of sector $= \frac{\theta}{360^\circ} \times \pi r^2$

Substitute the given values $r = 28 \text{ cm}$, $\theta = 60^\circ$, and $\pi = \frac{22}{7}$:

Simplify the fraction $\frac{60^\circ}{360^\circ}$:

Perform the calculation and cancel common factors:

Let's calculate more carefully:

2. Area of the triangle in one sector:

The triangle is formed by two radii (length $r = 28 \text{ cm}$) and the chord. The angle between the radii is $60^\circ$. An isosceles triangle with an included angle of $60^\circ$ is an equilateral triangle. The side length of this triangle is equal to the radius, $28 \text{ cm}$.

The area of an equilateral triangle with side $s$ is $\frac{\sqrt{3}}{4} s^2$. Using $s = r = 28 \text{ cm}$ and $\sqrt{3} = 1.7$:

Perform the calculation:

Calculate $1.7 \times 196$:

3. Area of one design (segment):

Area of one design $=$ Area of sector $-$ Area of triangle

Convert $333.2$ to a fraction: $333.2 = \frac{3332}{10} = \frac{1666}{5}$.

Find a common denominator (15):

Perform the subtraction:

4. Total area of the six designs:

Simplify the expression:

Convert to decimal:

5. Cost of making the designs:

Cost $=$ Total area of designs $\times$ Rate per $\text{cm}^2$

Calculate $464.8 \times 0.35$:

---

Final Answer:

The cost of making the designs is $\textsf{₹} 162.68$.

Given:

Square ABCD with side $s = 14 \text{ cm}$.

Four circles are drawn with centers A, B, C, and D, such that each circle touches externally two other circles.

Value of $\pi = \frac{22}{7}$.

---

To Find:

The area of the shaded region.

---

Solution:

Since each circle is drawn with a vertex of the square as its center and touches two other circles, the radius of each circle is half the side length of the square.

The shaded region is the area of the square minus the area of the four sectors (quarter circles) at the corners.

1. Area of the square:

Area of square $= \text{side}^2$

2. Area of the four sectors:

Each sector is a quarter circle with radius $r = 7 \text{ cm}$, centered at a vertex of the square (angle $90^\circ$).

The area of one sector is $\frac{1}{4}$ of the area of a full circle with the same radius, or using the sector formula with $\theta = 90^\circ$:

Area of one sector $= \frac{90^\circ}{360^\circ} \times \pi r^2 = \frac{1}{4} \pi r^2$

There are four such sectors. The total area of the four sectors is:

Substitute the values $r = 7 \text{ cm}$ and $\pi = \frac{22}{7}$:

Simplify the expression by cancelling common factors:

3. Area of the shaded region:

Area of shaded region $=$ Area of square $-$ Total area of four sectors

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Final Answer:

The area of the shaded region is $42 \text{ cm}^2$.

Given:

Square of side $s = 4 \text{ cm}$.

From each corner, a quadrant of a circle of radius $r_1 = 1 \text{ cm}$ is cut.

From the center, a circle of diameter $d_2 = 2 \text{ cm}$ is cut.

Value of $\pi = \frac{22}{7}$.

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To Find:

The area of the remaining portion of the square.

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Solution:

The area of the remaining portion of the square is the area of the original square minus the total area of the four quadrants and the area of the circle cut from the center.

1. Area of the square:

Area of square $= \text{side}^2$

2. Total area of the four quadrants:

From each of the four corners, a quadrant of a circle with radius $r_1 = 1 \text{ cm}$ is cut. Each quadrant is a sector with angle $90^\circ$.

The area of one quadrant is $\frac{1}{4}$ of the area of a full circle with radius $r_1$.

Area of one quadrant $= \frac{1}{4} \pi r_1^2$

The total area of the four quadrants is $4 \times \left(\frac{1}{4} \pi r_1^2\right) = \pi r_1^2$.

Substitute the values $r_1 = 1 \text{ cm}$ and $\pi = \frac{22}{7}$:

3. Area of the circle cut from the center:

A circle of diameter $d_2 = 2 \text{ cm}$ is cut from the center. The radius of this circle is $r_2 = \frac{d_2}{2} = \frac{2 \text{ cm}}{2} = 1 \text{ cm}$.

The area of this circle is $\pi r_2^2$.

Substitute the values $r_2 = 1 \text{ cm}$ and $\pi = \frac{22}{7}$:

4. Area of the remaining portion:

Area of remaining portion $=$ Area of square $-$ (Total area of four quadrants $+$ Area of center circle)

Combine the terms by finding a common denominator (7):

Perform the subtraction:

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Final Answer:

The area of the remaining portion of the square is $\frac{68}{7} \text{ cm}^2$.

Given:

Area of an equilateral triangle $= 49\sqrt{3} \text{ cm}^2$.

Three circles are drawn with centers at each vertex of the triangle.

The radius of each circle is half the length of the side of the triangle.

Value of $\sqrt{3} = 1.73$.

Value of $\pi = \frac{22}{7}$.

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To Find:

The area of the triangle not included in the circles.

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Solution:

Let the side length of the equilateral triangle be $a \text{ cm}$.

The area of an equilateral triangle with side $a$ is given by the formula: Area $= \frac{\sqrt{3}}{4} a^2$.

We are given the area is $49\sqrt{3} \text{ cm}^2$. Substitute this into the formula:

Divide both sides by $\sqrt{3}$ (assuming $\sqrt{3} \neq 0$):

Multiply both sides by 4:

Take the square root of both sides to find $a$. Since side length must be positive:

The side length of the equilateral triangle is $14 \text{ cm}$.

The radius of each circle drawn with the vertex as center is half the side length of the triangle.

At each vertex of an equilateral triangle, the angle is $60^\circ$. Thus, each circle cuts a sector of angle $60^\circ$ from the triangle.

The area of one such sector is given by the formula:

Area of sector $= \frac{\theta}{360^\circ} \times \pi r^2$

Substitute the values $r = 7 \text{ cm}$, $\theta = 60^\circ$, and $\pi = \frac{22}{7}$:

Perform the calculation and cancel common factors:

There are three such sectors within the triangle (one at each vertex). The total area of the three sectors within the triangle is:

Alternatively, three sectors of $60^\circ$ each combine to form a semicircle ($3 \times 60^\circ = 180^\circ$). However, since the circles are centered at the vertices and the radius is half the side, these sectors fit within the triangle. The total angle of these three sectors is $180^\circ$, covering half a circle of radius $7 \text{ cm}$. The area is $\frac{1}{2} \pi r^2 = \frac{1}{2} \times \frac{22}{7} \times 7^2 = \frac{1}{2} \times \frac{22}{7} \times 49 = 11 \times 7 = 77 \text{ cm}^2$. Both methods are consistent.

The area of the triangle not included in the circles is the area of the equilateral triangle minus the total area of the three sectors within the triangle.

Substitute the given value $\sqrt{3} = 1.73$ into the area of the triangle:

Calculate $49 \times 1.73$:

Now, perform the subtraction to find the area not included:

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Final Answer:

The area of the triangle not included in the circles is $7.77 \text{ cm}^2$.

Given:

Inner circumference of the race track, $C_1 = 352 \text{ m}$.

Outer circumference of the race track, $C_2 = 396 \text{ m}$.

The track is in the form of a ring.

Value of $\pi = \frac{22}{7}$.

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To Find:

1. The width of the track.

2. The area of the track.

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Solution:

Let the inner radius of the track be $r_1$ and the outer radius be $r_2$.

The formula for the circumference of a circle with radius $r$ is $C = 2\pi r$.

For the inner circle:

To find $r_1$, multiply both sides by $\frac{7}{44}$:

Simplify the expression:

For the outer circle:

To find $r_2$, multiply both sides by $\frac{7}{44}$:

Simplify the expression:

1. Width of the track:

The width of the track is the difference between the outer radius and the inner radius.

2. Area of the track:

The area of the track is the difference between the area of the outer circle and the area of the inner circle.

Area of track $= \pi r_2^2 - \pi r_1^2$

Factor out $\pi$:

Substitute the values $r_1 = 56 \text{ m}$, $r_2 = 63 \text{ m}$, and $\pi = \frac{22}{7}$:

Using the difference of squares formula, $a^2 - b^2 = (a-b)(a+b)$:

Simplify the expression by cancelling common factors:

Perform the multiplication:

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Final Answer:

The width of the track is $7 \text{ m}$.

The area of the track is $2618 \text{ m}^2$.

Given:

Square ABCD with side $s = 21 \text{ cm}$.

Two semicircles are drawn with BC and AD as diameters.

Value of $\pi = \frac{22}{7}$.

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To Find:

The area of the shaded region.

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Solution:

The shaded region is the area of the square from which the two semicircles are removed.

1. Area of the square:

Area of square $= \text{side}^2$

2. Area of the two semicircles:

The diameter of each semicircle is equal to the side of the square, $d = 21 \text{ cm}$.

The radius of each semicircle is $r = \frac{d}{2} = \frac{21}{2} \text{ cm}$.

The area of one semicircle is $\frac{1}{2} \pi r^2$.

The total area of the two semicircles is $2 \times \left(\frac{1}{2} \pi r^2\right) = \pi r^2$.

Substitute the values $r = \frac{21}{2} \text{ cm}$ and $\pi = \frac{22}{7}$:

Simplify the expression by cancelling common factors:

Calculate $11 \times 63$:

3. Area of the shaded region:

Area of shaded region $=$ Area of square $-$ Total area of two semicircles

Perform the subtraction:

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Final Answer:

The area of the shaded region is $267.75 \text{ cm}^2$.

Given:

Radius of the circle, $r = 15 \text{ cm}$.

Angle subtended by the chord at the center, $\theta = 60^\circ$.

Value of $\pi = 3.14$.

Value of $\sqrt{3} = 1.73$.

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To Find:

1. The area of the minor segment.

2. The area of the corresponding major segment.

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Solution:

1. Area of the minor segment:

The area of the minor segment is the difference between the area of the corresponding sector and the area of the triangle formed by the radii and the chord.

Area of sector $= \frac{\theta}{360^\circ} \times \pi r^2$

Substitute the given values $r = 15 \text{ cm}$, $\theta = 60^\circ$, and $\pi = 3.14$:

Calculate $3.14 \times 225$:

The triangle formed by the radii and the chord has sides $r$ and $r$ and the included angle $\theta$. Its area is given by:

Area of triangle $= \frac{1}{2} r^2 \sin(\theta)$

Substitute the values $r = 15 \text{ cm}$ and $\theta = 60^\circ$:

We know $\sin(60^\circ) = \frac{\sqrt{3}}{2}$. Use $\sqrt{3} = 1.73$:

Calculate $225 \times 1.73$:

Area of minor segment $=$ Area of minor sector $-$ Area of triangle

Perform the subtraction:

2. Area of the major segment:

The area of the major segment is the area of the full circle minus the area of the minor segment.

Area of circle $= \pi r^2$

Substitute the values $r = 15 \text{ cm}$ and $\pi = 3.14$:

Area of major segment $=$ Area of circle $-$ Area of minor segment

Perform the subtraction:

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Final Answer:

The area of the minor segment is $20.4375 \text{ cm}^2$.

The area of the major segment is $686.0625 \text{ cm}^2$.

Given:

ABC is a quadrant of a circle with center A and radius $r_1 = 14 \text{ cm}$.

A semicircle is drawn with BC as diameter.

Value of $\pi = \frac{22}{7}$.

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To Find:

The area of the shaded region.

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Solution:

The shaded region is the area of the semicircle drawn on BC as diameter, excluding the portion that overlaps with the quadrant.

The area of the shaded region can be found by subtracting the area of the segment formed by the chord BC within the quadrant from the area of the semicircle drawn on BC.

1. Area of the quadrant ABC:

The angle of the quadrant at the center A is $90^\circ$. The radius is $r_1 = 14 \text{ cm}$.

Area of quadrant $= \frac{90^\circ}{360^\circ} \times \pi r_1^2 = \frac{1}{4} \pi r_1^2$

Perform the calculation and cancel common factors:

2. Area of triangle ABC:

Triangle ABC is a right-angled triangle at A, with sides AB and AC as radii of the quadrant ($r_1 = 14 \text{ cm}$).

Area of triangle ABC $= \frac{1}{2} \times \text{base} \times \text{height}$

3. Area of the segment of the quadrant formed by chord BC:

Area of segment $=$ Area of quadrant ABC $-$ Area of triangle ABC

4. Area of the semicircle drawn on BC:

BC is the diameter of the semicircle. In right triangle ABC, BC is the hypotenuse.

Diameter of semicircle, $d_2 = 14\sqrt{2} \text{ cm}$.

Radius of semicircle, $r_2 = \frac{d_2}{2} = \frac{14\sqrt{2}}{2} = 7\sqrt{2} \text{ cm}$.

Area of semicircle $= \frac{1}{2} \pi r_2^2$

Perform the calculation and cancel common factors:

5. Area of the shaded region:

The shaded region is the area of the semicircle minus the area of the segment of the quadrant above the chord BC.

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Final Answer:

The area of the shaded region is $98 \text{ cm}^2$.

Given:

Radius of the pulley (circle), $r = OA = OB = 5 \text{ cm}$.

Distance from the center O to point P, $OP = 10 \text{ cm}$.

PA and PB are tangents to the circle from point P.

Value of $\pi = 3.14$.

Value of $\sqrt{3} = 1.73$.

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To Find:

1. The area of the quadrilateral OAPB.

2. The area of the shaded region (Area(OAPB) - Area(minor sector OAB)).

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Solution:

1. Area of the quadrilateral OAPB:

We know that the radius to the point of tangency is perpendicular to the tangent. Therefore, $\angle OAP = 90^\circ$ and $\angle OBP = 90^\circ$.

Consider the right-angled triangle $\triangle OAP$. We have OA (radius) and OP (distance from center to P).

Using the Pythagorean theorem in $\triangle OAP$:

The quadrilateral OAPB can be divided into two right-angled triangles, $\triangle OAP$ and $\triangle OBP$. These triangles are congruent.

Area of right-angled triangle $\triangle OAP = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times \text{OA} \times \text{PA}$.

Using the given value $\sqrt{3} = 1.73$:

The area of the quadrilateral OAPB is twice the area of $\triangle OAP$:

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2. Area of the shaded region:

The shaded region is the area of the quadrilateral OAPB minus the area of the minor sector OAB. To find the area of the sector, we need the angle $\angle AOB$.

In right-angled triangle $\triangle OAP$, we can find the angle $\angle AOP$.

The angle whose cosine is $\frac{1}{2}$ is $60^\circ$.

Since $\triangle OAP \cong \triangle OBP$, $\angle AOP = \angle BOP$.

The area of the minor sector OAB with angle $\theta = 120^\circ$ and radius $r = 5 \text{ cm}$ is:

Area of sector OAB $= \frac{\theta}{360^\circ} \times \pi r^2$

Convert the fraction to a decimal for subtraction:

The area of the shaded region is the area of quadrilateral OAPB minus the area of the minor sector OAB.

To perform the subtraction, use the decimal value of the sector area (rounded for calculation):

Perform the subtraction:

Rounding to two decimal places, the area of the shaded region is $17.08 \text{ cm}^2$.

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Final Answer:

The area of the quadrilateral OAPB is $43.25 \text{ cm}^2$.

The area of the shaded region is approximately $17.08 \text{ cm}^2$.

Given:

Inner dimensions of the rectangular park: Length ($l_1$) = $120 \text{ m}$, Width ($w_1$) = $100 \text{ m}$.

Diameter of the circular fountain = $14 \text{ m}$.

Width of the rectangular path along the sides = $2 \text{ m}$.

Width of the circular path around the fountain = $3 \text{ m}$.

Value of $\pi = \frac{22}{7}$.

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To Find:

The total area of the paths.

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Solution:

The total area of the paths consists of two parts: the rectangular path along the sides of the park and the circular path around the fountain.

1. Area of the rectangular path:

The rectangular path is $2 \text{ m}$ wide and runs along the sides of the $120 \text{ m} \times 100 \text{ m}$ park. We assume this path is outside the park, forming a larger rectangle.

Inner length of the rectangular park, $l_1 = 120 \text{ m}$.

Inner width of the rectangular park, $w_1 = 100 \text{ m}$.

Area of the inner rectangular park, $A_1 = l_1 \times w_1 = 120 \text{ m} \times 100 \text{ m} = 12000 \text{ m}^2$.

The rectangular path of width $2 \text{ m}$ adds $2 \text{ m}$ to each side of the length and width.

Outer length of the park including the path, $l_2 = l_1 + 2 \times (\text{path width}) = 120 \text{ m} + 2 \times 2 \text{ m} = 120 + 4 = 124 \text{ m}$.

Outer width of the park including the path, $w_2 = w_1 + 2 \times (\text{path width}) = 100 \text{ m} + 2 \times 2 \text{ m} = 100 + 4 = 104 \text{ m}$.

Area of the outer rectangle including the path, $A_2 = l_2 \times w_2 = 124 \text{ m} \times 104 \text{ m}$.

Area of the rectangular path, $A_{\text{rect path}} = A_2 - A_1$.

2. Area of the circular path around the fountain:

The fountain is circular with diameter $14 \text{ m}$.

Radius of the fountain (inner circle of the path), $r_f = \frac{\text{Diameter}}{2} = \frac{14 \text{ m}}{2} = 7 \text{ m}$.

The circular path is $3 \text{ m}$ wide and runs around the fountain.

Radius of the fountain plus the path (outer circle of the path), $r_p = r_f + \text{path width} = 7 \text{ m} + 3 \text{ m} = 10 \text{ m}$.

The area of the circular path is the difference between the area of the outer circle and the area of the inner circle (fountain).

Area of circular path, $A_{\text{circ path}} = \pi r_p^2 - \pi r_f^2 = \pi (r_p^2 - r_f^2)$

Substitute the values $r_f = 7 \text{ m}$, $r_p = 10 \text{ m}$, and $\pi = \frac{22}{7}$:

3. Total area of the paths:

Total area of paths $=$ Area of rectangular path $+$ Area of circular path

To add these values, express $896$ as a fraction with denominator 7:

Expressing as a decimal:

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Final Answer:

The area of the paths is $\frac{7394}{7} \text{ m}^2$ or approximately $1056.29 \text{ m}^2$ (rounded to two decimal places).