Chapter 14 Symmetry (Additional Questions)

The correct option is:

(C) Mirror

Explanation:

Line symmetry is defined as the property where a figure can be folded along a line (the line of symmetry) such that one half is the exact reflection of the other half.

This type of reflection across a line is commonly referred to as a mirror reflection because it behaves like reflecting the figure in a mirror placed along the line of symmetry.

Rotational symmetry involves rotation around a point.

Translational symmetry involves shifting the figure without rotation or reflection.

Enlarged reflections would scale the figure, which is not part of the definition of line symmetry where the halves are identical.

The correct option is:

(B) Line of symmetry

Explanation:

The definition of line symmetry states that a figure is symmetrical if it can be divided by a line into two parts that are mirror images of each other.

This dividing line is specifically called the line of symmetry.

The axis of rotation is related to rotational symmetry.

A diagonal is a line segment connecting two non-adjacent vertices of a polygon.

A transversal is a line that intersects two or more other lines at distinct points.

The correct option is:

(A) A

Explanation:

Let's examine the symmetry of each letter given in the options:

(A) A: This letter has a vertical line of symmetry that passes through the middle of the letter, splitting it into two identical mirror halves.

(B) B: This letter has a horizontal line of symmetry, splitting it into two identical mirror halves. It does not have vertical symmetry.

(C) C: This letter has a horizontal line of symmetry, splitting it into two identical mirror halves. It does not have vertical symmetry.

(D) E: This letter has a horizontal line of symmetry, splitting it into two identical mirror halves. It does not have vertical symmetry.

Therefore, only letter 'A' among the given options has exactly one line of vertical symmetry.

A line of symmetry is a line that divides a figure into two identical halves that are mirror images of each other.

For a rectangle, there are two such lines:

1. The line joining the midpoints of the longer sides.

2. The line joining the midpoints of the shorter sides.

Folding the rectangle along either of these lines makes the two halves coincide exactly.

Therefore, a rectangle has 2 lines of symmetry.

The correct option is (C) 2.

A line of symmetry divides a figure into two mirror-image halves.

An equilateral triangle has three equal sides and three equal angles ($60^\circ$). Due to its perfect symmetry, it has multiple lines of symmetry.

The lines of symmetry of an equilateral triangle are the lines that pass through each vertex and the midpoint of the opposite side. These lines are also the altitudes, medians, and angle bisectors of the triangle.

Since there are three vertices, there are three such lines.

Therefore, an equilateral triangle has 3 lines of symmetry.

The correct option is (C) 3.

A line of symmetry divides a figure into two identical mirror halves.

For a circle, any line that passes through its center is a line of symmetry. If you fold the circle along any diameter (which is a line passing through the center), the two halves will perfectly overlap.

Since there are an infinite number of diameters that can be drawn through the center of a circle, there are an infinite number of lines of symmetry.

Therefore, a circle has infinitely many lines of symmetry.

The correct option is (D) Infinitely many.

Let's examine the lines of symmetry for each figure:

(A) Isosceles triangle: An isosceles triangle has two equal sides and two equal angles. It has one line of symmetry, which passes through the vertex angle and the midpoint of the base (or is the altitude from the vertex angle to the base).

(B) Scalene triangle: A scalene triangle has all sides of different lengths and all angles of different measures. There is no line that can divide a scalene triangle into two congruent mirror-image halves.

(C) Square: A square is a highly symmetric figure. It has four lines of symmetry: the two diagonals and the two lines joining the midpoints of opposite sides.

(D) Rhombus: A rhombus is a quadrilateral with all four sides of equal length. It has two lines of symmetry, which are its diagonals.

Based on the analysis above, the figure with no line of symmetry is the scalene triangle.

The correct option is (B) Scalene triangle.

Rotational symmetry means that a figure can be rotated about a central point by some angle less than $360^\circ$ and still look the same.

The angle of rotational symmetry is the smallest angle of rotation for which the figure coincides with its original position.

A square has a center point. If we rotate a square about its center by $90^\circ$, the square maps onto itself. Similarly, rotations by $180^\circ$, $270^\circ$, and $360^\circ$ also map the square onto itself.

The smallest of these angles (excluding $360^\circ$ which is always a symmetry for any figure) is $90^\circ$.

Therefore, the angle of rotational symmetry for a square is $90^\circ$.

The correct option is (A) $90^\circ$.

Rotational symmetry is when a figure can be rotated around a central point by some angle and still look the same.

The order of rotational symmetry is the number of times a figure coincides with itself during a rotation of $360^\circ$. It can be calculated as $\frac{360^\circ}{\text{angle of rotational symmetry}}$.

An equilateral triangle has three equal sides and three equal angles ($60^\circ$). Its center of rotation is the centroid (the intersection of medians).

If we rotate an equilateral triangle about its center, it will coincide with its original position after rotations of $120^\circ$, $240^\circ$, and $360^\circ$.

The smallest angle of rotational symmetry is $120^\circ$.

The order of rotational symmetry is the number of times it coincides within $360^\circ$. It coincides at $120^\circ$, $240^\circ$, and $360^\circ$. That's 3 times.

Alternatively, Order = $\frac{360^\circ}{120^\circ} = 3$.

Therefore, the order of rotational symmetry for an equilateral triangle is 3.

The correct option is (C) 3.

The center of rotation is the fixed point about which a figure rotates. For a figure to have rotational symmetry, it must coincide with its original position after rotation by an angle less than $360^\circ$ around this center point.

In a square, the point around which rotations of $90^\circ$, $180^\circ$, and $270^\circ$ map the square onto itself is the geometric center of the square.

Let's consider the given options:

(A) Sides: Sides are line segments, not a single point.

(B) Vertices: There are four vertices. Rotating around a single vertex would not make the square coincide with itself unless the rotation is $360^\circ$.

(C) Diagonals: The two diagonals of a square intersect at a single point at the center of the square. This point is equidistant from all vertices and lies on all lines of symmetry. Rotating the square around this point results in rotational symmetry.

(D) Medians: In the context of quadrilaterals, medians usually refer to line segments connecting the midpoints of opposite sides. These lines also intersect at the center of the square. However, the intersection of diagonals is the standard way to define the center of a square.

The point of intersection of the diagonals of a square is its geometric center, and this point serves as its center of rotation.

The correct option is (C) Diagonals.

A figure has rotational symmetry if, when rotated by a certain angle less than $360^\circ$ about a fixed point (the center of rotation), it coincides with its original position.

Let $\theta$ be an angle of rotational symmetry. If we rotate the figure by $\theta$, it looks the same. If we rotate it by $\theta$ again (a total of $2\theta$), it still looks the same. This is true for any multiple of $\theta$, i.e., $k\theta$ for $k=1, 2, 3, \ldots$.

After a rotation of $360^\circ$, any figure will return to its original position. Therefore, a rotation by $360^\circ$ is always a rotational symmetry.

For a figure with rotational symmetry by an angle $\theta$, the figure must also coincide with its original position after a $360^\circ$ rotation. This means that $360^\circ$ must be a multiple of $\theta$. In other words, the angle of rotation $\theta$ must be a factor (divisor) of $360^\circ$.

Let's say the smallest angle of rotational symmetry is $\theta_{min}$. Then the angles of rotational symmetry are $\theta_{min}, 2\theta_{min}, 3\theta_{min}, \ldots$. One of these angles must be $360^\circ$. So, $k \cdot \theta_{min} = 360^\circ$ for some positive integer $k$. This shows that $\theta_{min}$ is a factor of $360^\circ$. Any other angle of rotational symmetry, being a multiple of $\theta_{min}$, will also be a factor of $360^\circ$.

Therefore, the angle of rotation must be a factor of $360^\circ$.

The correct option is (D) $360^\circ$.

Rotational symmetry is when a figure coincides with its original position after rotation by a certain angle less than $360^\circ$ about a fixed point (the center of rotation).

The center of rotation for a parallelogram is the point of intersection of its diagonals.

When a parallelogram is rotated by $180^\circ$ about the intersection of its diagonals, each vertex maps to the opposite vertex (e.g., A to C, B to D). Since opposite sides are equal and parallel, the rotated figure exactly coincides with the original figure.

The smallest angle of rotation (greater than $0^\circ$) for which a parallelogram coincides with itself is $180^\circ$.

The order of rotational symmetry is the number of times the figure coincides with itself during a $360^\circ$ rotation. It is calculated as $\frac{360^\circ}{\text{smallest angle of rotational symmetry}}$.

Order = $\frac{360^\circ}{180^\circ} = 2$.

This holds true for all parallelograms, including rectangles and rhombuses. The exclusion in the question clarifies that we are looking for the property common to all parallelograms, which is $180^\circ$ rotational symmetry.

Therefore, the order of rotational symmetry for a parallelogram is 2.

The correct option is (B) 2.

We need to find the figure that has rotational symmetry with an order greater than 1 and simultaneously has no line symmetry.

Let's examine each option:

(A) Rectangle: A rectangle has rotational symmetry of order 2 (it coincides with itself after a $180^\circ$ rotation). It also has 2 lines of symmetry (joining the midpoints of opposite sides).

(B) Parallelogram (not rhombus/rectangle): A parallelogram has rotational symmetry of order 2 (it coincides with itself after a $180^\circ$ rotation about the intersection of its diagonals). A parallelogram that is not a rhombus (sides are not all equal) and not a rectangle (angles are not $90^\circ$) does not have any line of symmetry.

(C) Circle: A circle has rotational symmetry of infinite order (it coincides with itself after rotation by any angle about its center). It also has infinitely many lines of symmetry (any line passing through the center).

(D) Isosceles triangle: A non-equilateral isosceles triangle has rotational symmetry of order 1 (only coincides with itself after a $360^\circ$ rotation). It has 1 line of symmetry.

Comparing the properties with the requirements:

• Rectangle: Order of rotation > 1, but has line symmetry.
• Parallelogram (not rhombus/rectangle): Order of rotation > 1, and has no line symmetry.
• Circle: Order of rotation > 1, but has line symmetry.
• Isosceles triangle: Order of rotation is 1 (not greater than 1) and has line symmetry.

Only the parallelogram (which is not a rhombus or rectangle) satisfies both conditions.

The correct option is (B) Parallelogram (not rhombus/rectangle).

Rotational symmetry of order $n$ means that a figure coincides with its original position exactly $n$ times when rotated by $360^\circ$ about its center.

The smallest angle of rotational symmetry for a figure with order $n$ is $\frac{360^\circ}{n}$.

If the order of rotational symmetry is 1 ($n=1$), the smallest angle of rotational symmetry is $\frac{360^\circ}{1} = 360^\circ$.

A rotation by $360^\circ$ always brings any figure back to its original position. Therefore, having rotational symmetry of order 1 means the only angle of rotation (less than or equal to $360^\circ$) for which the figure coincides with its original position is $360^\circ$.

This is equivalent to saying that the figure has no rotational symmetry for any angle between $0^\circ$ and $360^\circ$. In common usage, figures with order 1 rotational symmetry are considered to have no rotational symmetry (other than the trivial identity rotation of $0^\circ$ or $360^\circ$).

Thus, a figure with rotational symmetry of order 1 has no rotational symmetry other than the rotation by $360^\circ$.

The correct option is (A) It has no rotational symmetry (other than a rotation by $360^\circ$).

A line of symmetry is a line that divides a figure into two congruent halves that are mirror images of each other.

A rhombus is a quadrilateral with all four sides of equal length. Its opposite angles are equal.

For a rhombus, the lines of symmetry are its diagonals. If you fold the rhombus along either of its diagonals, the two resulting triangles are congruent and will perfectly overlap.

A rhombus has two diagonals.

Therefore, a rhombus has 2 lines of symmetry.

The correct option is (C) 2.

We are looking for a figure that satisfies two conditions:

1. It has exactly two lines of symmetry.

2. It has rotational symmetry of order 2.

Let's examine the properties of each option:

(A) Square: A square has 4 lines of symmetry (two diagonals and two lines joining midpoints of opposite sides). It has rotational symmetry of order 4 (coincides after $90^\circ, 180^\circ, 270^\circ, 360^\circ$ rotations). This does not fit the criteria.

(B) Equilateral triangle: An equilateral triangle has 3 lines of symmetry (from each vertex to the midpoint of the opposite side). It has rotational symmetry of order 3 (coincides after $120^\circ, 240^\circ, 360^\circ$ rotations). This does not fit the criteria.

(C) Rectangle (not square): A rectangle that is not a square has unequal adjacent sides. It has exactly two lines of symmetry: the lines joining the midpoints of opposite sides. It has rotational symmetry of order 2, as it coincides with its original position after a $180^\circ$ rotation about the intersection of its diagonals. This fits both criteria.

(D) Rhombus (not square): A rhombus that is not a square has unequal angles (not $90^\circ$). It has exactly two lines of symmetry: its diagonals. It has rotational symmetry of order 2, as it coincides with its original position after a $180^\circ$ rotation about the intersection of its diagonals. This also fits both criteria.

Both option (C) and option (D) satisfy the given conditions based on standard geometric definitions. However, in a multiple-choice question with a single correct answer, there might be a preference for one figure over the other in the context from which the question is taken. Assuming there is a unique intended answer among the choices, and given that a rectangle is often used as a primary example of a figure with exactly two lines of symmetry and order 2 rotational symmetry, we select (C).

The correct option is likely (C) Rectangle (not square), although (D) also fits the description.

A regular pentagon is a polygon with 5 equal sides and 5 equal interior angles. It has rotational symmetry about its center.

The order of rotational symmetry for a regular $n$-sided polygon is $n$. For a regular pentagon, the order is 5.

The angle of rotational symmetry is the smallest angle through which the figure can be rotated about its center to coincide with its original position. This angle is calculated as $\frac{360^\circ}{\text{order of rotational symmetry}}$.

For a regular pentagon, the angle of rotational symmetry is $\frac{360^\circ}{5}$.

Calculation:

$\frac{360}{5} = 72$

So, the angle of rotational symmetry is $72^\circ$.

Therefore, the angle of rotational symmetry for a regular pentagon is $72^\circ$.

The correct option is (B) $72^\circ$.

We need to identify the letters of the English alphabet that possess both a horizontal line of symmetry and a vertical line of symmetry.

A horizontal line of symmetry means the letter can be folded in half horizontally and the top half matches the bottom half.

A vertical line of symmetry means the letter can be folded in half vertically and the left half matches the right half.

Let's check the symmetry for common uppercase letters:

• A: Vertical only.
• B, C, D, E: Horizontal only.
• F, G: No line symmetry.
• H: Both horizontal and vertical.
• I: Both horizontal and vertical.
• J: No line symmetry.
• K: Horizontal only.
• L: No line symmetry.
• M: Vertical only.
• N: No line symmetry (has rotational symmetry of order 2).
• O: Both horizontal and vertical (and infinite rotational symmetry).
• P, Q, R: No line symmetry.
• S: No line symmetry (has rotational symmetry of order 2).
• T: Vertical only.
• U: Vertical only.
• V: Vertical only.
• W: Vertical only.
• X: Both horizontal and vertical.
• Y: Vertical only.
• Z: No line symmetry (has rotational symmetry of order 2).

The letters that have both horizontal and vertical lines of symmetry are H, I, O, and X.

Now let's look at the options provided:

• (A) A, M, T, U, V, W, Y: These letters generally only have vertical symmetry.
• (B) B, C, D, E, K: These letters generally only have horizontal symmetry.
• (C) H, I, O, X: All these letters have both horizontal and vertical symmetry.
• (D) F, G, J, L, N, P, Q, R, S, Z: These letters generally have no line symmetry.

Therefore, the set of letters that have both horizontal and vertical lines of symmetry is H, I, O, X.

The correct option is (C) H, I, O, X.

Rotational symmetry occurs when a figure can be rotated about a fixed point (its center of rotation) by an angle less than $360^\circ$ and coincide with its original position.

The angle of rotation (specifically, the smallest angle of rotation greater than $0^\circ$) describes the minimum rotation required for the figure to appear unchanged. This angle is a fundamental characteristic of the rotational symmetry.

The order of rotational symmetry is the number of times the figure coincides with its original position during a full $360^\circ$ rotation. It is related to the smallest angle of rotation ($\theta_{min}$) by the formula: Order $= \frac{360^\circ}{\theta_{min}}$.

Both the angle of rotation and the order of rotation are used to describe the nature and extent of a figure's rotational symmetry.

Since both the order of rotation and the angle of rotation are used to describe rotational symmetry, the most comprehensive answer is that it is described by both.

The correct option is (D) Both (B) and (C).

A line of symmetry is a line that divides a figure into two congruent mirror-image halves.

A semi-circle consists of a curved arc and a straight line segment (the diameter).

Let's consider potential lines of symmetry:

1. The diameter itself: This is the boundary, not a line through the figure that can be folded upon.

2. Lines perpendicular to the diameter: Only one such line will divide the semi-circle into two identical halves. This line is the perpendicular bisector of the diameter. It passes through the midpoint of the diameter (which is the center of the original circle) and is perpendicular to the diameter. Folding the semi-circle along this line makes the curved arc map onto itself and the diameter segment map onto itself.

3. Lines at other angles: Any other line passing through the center or elsewhere will not divide the semi-circle into two mirror images.

Therefore, a semi-circle has exactly 1 line of symmetry.

The correct option is (B) 1.

Let's examine each statement:

(A) A figure with line symmetry must also have rotational symmetry.

Consider an isosceles triangle that is not equilateral. It has one line of symmetry (the altitude from the vertex angle to the base). However, its rotational symmetry is of order 1, meaning it only coincides with itself after a $360^\circ$ rotation. This is not considered rotational symmetry (other than the trivial case).

Therefore, this statement is FALSE.

(B) A figure with rotational symmetry must also have line symmetry.

Consider a parallelogram that is not a rectangle or a rhombus. It has rotational symmetry of order 2 (it coincides with its original position after a $180^\circ$ rotation about the intersection of its diagonals). However, such a parallelogram has no lines of symmetry.

Therefore, this statement is FALSE.

(C) A figure can have rotational symmetry without line symmetry.

As shown in the analysis of statement (B), a parallelogram (that is not a rectangle or rhombus) is a figure that has rotational symmetry (order 2) but no line symmetry.

Therefore, this statement is TRUE.

(D) A figure can have more lines of symmetry than its order of rotational symmetry.

For common geometric figures with finite symmetry, the number of lines of symmetry ($L$) is often equal to the order of rotational symmetry ($R$) (e.g., squares, regular polygons, rectangles, rhombuses). In some cases, like a parallelogram (not rectangle/rhombus), $L=0$ and $R=2$, so $L < R$. For bounded figures with point symmetry, the number of lines of symmetry is generally less than or equal to the order of rotational symmetry ($L \leq R$), with equality holding for dihedral symmetry and $L=0$ for cyclic symmetry with order greater than 1.

Therefore, this statement is generally FALSE in the context of typical figures studied in this topic, and counterexamples where $L>R$ are not readily found among standard shapes with finite point symmetry.

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

The correct option is (C) A figure can have rotational symmetry without line symmetry.

We need to determine the number of lines of symmetry for each figure listed:

(i) Square: A square has 4 lines of symmetry. These are the two diagonals and the two lines joining the midpoints of opposite sides.

(ii) Circle: A circle has infinitely many lines of symmetry. Any line passing through the center of the circle is a line of symmetry (a diameter).

(iii) Isosceles triangle (non-equilateral): An isosceles triangle (which is not equilateral) has exactly 1 line of symmetry. This line is the altitude from the vertex angle to the base, which is also the angle bisector of the vertex angle and the median to the base.

(iv) Parallelogram (non-rhombus/rectangle): A parallelogram that is neither a rhombus nor a rectangle has no line of symmetry. It has rotational symmetry of order 2, but no line that divides it into mirror-image halves.

Matching the figures with the number of lines of symmetry:

(i) Square $\rightarrow$ 4 lines of symmetry (b)

(ii) Circle $\rightarrow$ Infinitely many lines of symmetry (d)

(iii) Isosceles triangle (non-equilateral) $\rightarrow$ 1 line of symmetry (a)

(iv) Parallelogram (non-rhombus/rectangle) $\rightarrow$ 0 lines of symmetry (c)

The correct matching is (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c).

Checking the options:

(A) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c) - This matches our result.

(B) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c) - Incorrect match for (ii) and (iii).

(C) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c) - Incorrect match for (i) and (ii).

(D) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a) - Incorrect match for (iii) and (iv).

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

The order of rotational symmetry of a figure is the number of times it coincides with itself during a complete rotation of $360^\circ$ about its center.

The order of rotational symmetry for a regular $n$-sided polygon is $n$.

Let's determine the order of rotational symmetry for each figure:

(i) Equilateral triangle: An equilateral triangle is a regular 3-sided polygon. Its order of rotational symmetry is 3. This matches option (c).

(ii) Square: A square is a regular 4-sided polygon. Its order of rotational symmetry is 4. This matches option (b).

(iii) Rectangle (non-square): A rectangle (that is not a square) coincides with itself after a rotation of $180^\circ$ about the intersection of its diagonals. The angles of rotation are $180^\circ$ and $360^\circ$. The order of rotational symmetry is $\frac{360^\circ}{180^\circ} = 2$. This matches option (a).

(iv) Regular hexagon: A regular hexagon is a regular 6-sided polygon. Its order of rotational symmetry is 6. This matches option (d).

The matching is:

(i) Equilateral triangle $\rightarrow$ Order 3 (c)

(ii) Square $\rightarrow$ Order 4 (b)

(iii) Rectangle (non-square) $\rightarrow$ Order 2 (a)

(iv) Regular hexagon $\rightarrow$ Order 6 (d)

Comparing this matching with the given options, we find that option (A) is correct:

(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)

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

Let's evaluate the Assertion (A) and the Reason (R).

Assertion (A): A square has 4 lines of symmetry and rotational symmetry of order 4.

A square is a quadrilateral with four equal sides and four right angles. It has four lines of symmetry: the two diagonals and the two lines joining the midpoints of opposite sides. The smallest angle of rotational symmetry for a square is $90^\circ$ ($360^\circ / 4$), and the order of rotational symmetry is $\frac{360^\circ}{90^\circ} = 4$. So, Assertion (A) is True.

Reason (R): A regular polygon with $n$ sides has $n$ lines of symmetry and rotational symmetry of order $n$.

For a regular polygon with $n$ sides, it is true that it has $n$ lines of symmetry. The lines of symmetry pass through vertices and midpoints of opposite sides (or midpoints of opposite sides, or opposite vertices, depending on whether $n$ is odd or even). It is also true that a regular $n$-sided polygon has rotational symmetry of order $n$, with the smallest angle of rotation being $\frac{360^\circ}{n}$. So, Reason (R) is True.

Now, let's determine if Reason (R) is the correct explanation for Assertion (A).

Assertion (A) makes a statement about a square, which is a specific type of regular polygon (with $n=4$). Reason (R) states a general property that applies to all regular polygons, including the square. Since a square is a regular polygon with $n=4$, applying the property from Reason (R) with $n=4$ gives: "A regular polygon with 4 sides has 4 lines of symmetry and rotational symmetry of order 4". This is exactly what Assertion (A) states.

Thus, Reason (R) is a general rule that directly explains why Assertion (A) is true for the specific case of a square.

Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

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

Let's evaluate the Assertion (A) and the Reason (R).

Assertion (A): The letter 'S' has line symmetry.

Consider the uppercase letter 'S'. A figure has line symmetry if it can be folded along a line such that the two halves are mirror images. The letter 'S' cannot be folded horizontally, vertically, or diagonally to produce mirror-image halves.

Therefore, Assertion (A) is False.

Reason (R): The letter 'S' has rotational symmetry of order 2.

A figure has rotational symmetry if it coincides with its original position after being rotated by an angle less than $360^\circ$ about a fixed point (its center). The order of rotational symmetry is the number of times it coincides during a $360^\circ$ rotation.

If the uppercase letter 'S' is rotated by $180^\circ$ about its center, it looks exactly the same as the original letter. This means it has rotational symmetry with an angle of $180^\circ$. The order of rotational symmetry is $\frac{360^\circ}{180^\circ} = 2$.

Therefore, Reason (R) is True.

We have found that Assertion (A) is False and Reason (R) is True.

Let's check the given options:

• (A) Both A and R are true... - Incorrect (A is false).
• (B) Both A and R are true... - Incorrect (A is false).
• (C) A is true, but R is false. - Incorrect (A is false and R is true).
• (D) A is false, but R is true. - Correct.

The correct option is (D) A is false, but R is true.

A line of symmetry is a line that divides a figure into two identical mirror-image halves.

Let's examine the Indian Rupee symbol $\textsf{₹}$ for possible lines of symmetry:

Horizontal symmetry: If we draw a horizontal line through the symbol, the upper part and the lower part are not mirror images of each other. The curved part and the bars do not align symmetrically.

Vertical symmetry: If we draw a vertical line through the symbol, the left part and the right part are not mirror images of each other. The left side is relatively open while the right side has the curved structure.

There are no other lines (like diagonals) that would provide symmetry for this symbol.

Since there is no line that can divide the Indian Rupee symbol $\textsf{₹}$ into two mirror-image halves, it has 0 lines of symmetry.

The correct option is (A) 0.

Rotational symmetry exists when a figure coincides with its original position after being rotated by an angle less than $360^\circ$ about a fixed point (its center).

The order of rotational symmetry is the number of times the figure coincides with itself during a complete rotation of $360^\circ$. An order of 1 means the figure only looks the same at the $360^\circ$ rotation (or $0^\circ$), which implies no rotational symmetry in the usual sense.

Let's consider the Indian Rupee symbol $\textsf{₹}$ and rotate it about its center.

A rotation by $180^\circ$ does not make the symbol look the same. The horizontal bars and the curve will be in different positions relative to the vertical stroke.

Rotating the symbol by any angle other than a multiple of $360^\circ$ will not result in the symbol coinciding with its original appearance.

Therefore, the only rotation (within $360^\circ$) that brings the symbol back to its original position is the $360^\circ$ rotation itself.

This means the order of rotational symmetry is 1.

The correct option is (B) 1.

A figure has a vertical line of symmetry if it can be folded in half vertically along a line, and the left half is a mirror image of the right half.

Let's examine the standard printed digits provided in the options for vertical symmetry:

(A) 0: The digit '0' has a vertical line of symmetry passing through its center. It also has a horizontal line of symmetry.

(B) 1: The digit '1' (often written as a single vertical stroke) has a vertical line of symmetry passing through its center. It does not have a horizontal line of symmetry.

(C) 8: The digit '8' has a vertical line of symmetry passing through its center. It also has a horizontal line of symmetry.

We are looking for the number that has exactly one line of vertical symmetry.

The digit '0' has one vertical line and one horizontal line (total 2 lines). It does not have *exactly one* vertical line.

The digit '1' has exactly one vertical line of symmetry and no horizontal line of symmetry.

The digit '8' has one vertical line and one horizontal line (total 2 lines). It does not have *exactly one* vertical line.

Therefore, the number that has exactly one line of vertical symmetry is 1.

The correct option is (B) 1.

We need to find a figure from the options that possesses rotational symmetry (order greater than 1) but does not have any line of symmetry.

Let's examine the symmetry properties of each figure:

(A) Square: A square has 4 lines of symmetry and rotational symmetry of order 4. It has both types of symmetry.

(B) Rhombus: A rhombus has 2 lines of symmetry (its diagonals) and rotational symmetry of order 2. It has both types of symmetry.

(C) Parallelogram (not special types): A parallelogram that is neither a rectangle nor a rhombus (i.e., its angles are not $90^\circ$ and its adjacent sides are not equal). It has rotational symmetry of order 2 about the intersection of its diagonals (it coincides with itself after a $180^\circ$ rotation). However, it does not have any line of symmetry.

(D) Equilateral triangle: An equilateral triangle has 3 lines of symmetry and rotational symmetry of order 3. It has both types of symmetry.

Comparing the properties with the requirement (rotational symmetry > order 1 AND no line symmetry), the parallelogram (not a special type) is the only figure among the options that fits this description.

The correct option is (C) Parallelogram (not special types).

The statement provides definitions for the angle of rotational symmetry and the order of rotational symmetry, and describes the relationship between them.

The "angle of rotation" is indeed defined as the smallest angle (greater than $0^\circ$) through which a figure can be rotated about its center to match its original position.

The "order of rotational symmetry" is the number of times a figure coincides with itself during a full rotation of $360^\circ$. If the smallest angle of rotational symmetry is $\theta_{min}$, then the figure will coincide with its original position after rotations of $\theta_{min}, 2\theta_{min}, 3\theta_{min}, \ldots$ until $n\theta_{min} = 360^\circ$, where $n$ is the total number of such positions (including the initial position) within $360^\circ$.

From $n\theta_{min} = 360^\circ$, we can express the order $n$ as:

$n = \frac{360^\circ}{\theta_{min}}$

This formula is a fundamental definition and relationship that holds true for any figure that possesses rotational symmetry (where $\theta_{min} < 360^\circ$, meaning order $n > 1$). Even for figures with no rotational symmetry (order 1), the smallest angle of coincidence is $360^\circ$, and the formula gives order $n = \frac{360^\circ}{360^\circ} = 1$.

This relationship is not limited to just regular polygons or circles; it applies to any figure that has rotational symmetry, such as parallelograms, certain letters of the alphabet (like H, I, O, X, N, S, Z), etc.

Therefore, the statement is always true as it defines the relationship between the smallest angle of rotational symmetry and the order of rotational symmetry for any figure.

The correct option is (A) Always true.

The statement describes a type of symmetry where a figure is divided into two halves that are mirror images of each other by a specific line.

Let's consider the definitions of the given types of symmetry:

• Rotational symmetry: A figure has rotational symmetry if it coincides with its original position after being rotated by an angle less than $360^\circ$ about a fixed point.
• Point symmetry: A figure has point symmetry if it coincides with its original position after a $180^\circ$ rotation about a point. It is a type of rotational symmetry.
• Line symmetry: A figure has line symmetry (also called reflectional symmetry) if there exists a line such that if the figure is folded along this line, the two halves coincide perfectly. The two halves are mirror images of each other with respect to this line.
• Translation symmetry: A figure has translation symmetry if it can be moved a certain distance in a certain direction and appear unchanged.

The description in the question "if there is a line that divides the figure into two mirror images" exactly matches the definition of line symmetry.

Therefore, the statement should be completed as: A figure has Line symmetry if there is a line that divides the figure into two mirror images.

The correct option is (C) Line.

A line of symmetry divides a figure into two identical mirror-image halves.

Let's determine the number of lines of symmetry for each of the given figures:

(A) Square: A square has 4 lines of symmetry: the two diagonals and the two lines joining the midpoints of opposite sides.

(B) Rectangle: A rectangle (that is not a square) has 2 lines of symmetry: the lines joining the midpoints of opposite sides.

(C) Circle: Any line passing through the center of a circle (i.e., any diameter) divides the circle into two congruent semicircles that are mirror images. Since there are infinitely many diameters that can be drawn through the center of a circle, a circle has infinitely many lines of symmetry.

(D) Equilateral triangle: An equilateral triangle has 3 lines of symmetry, each passing through a vertex and the midpoint of the opposite side.

Comparing the numbers, only the circle has an infinite number of lines of symmetry.

The correct option is (C) Circle.

The order of rotational symmetry of a figure is the number of times it coincides with its original position during a complete rotation of $360^\circ$ about its center.

For a regular polygon with $n$ sides, the order of rotational symmetry is equal to $n$. This is because a regular $n$-gon can be rotated by $\frac{360^\circ}{n}$ and it will coincide with itself. This smallest angle occurs $n$ times in a full $360^\circ$ rotation.

A regular hexagon is a polygon with 6 equal sides.

Using the property that the order of rotational symmetry for a regular polygon with $n$ sides is $n$, the order for a regular hexagon (with $n=6$) is 6.

The smallest angle of rotational symmetry is $\frac{360^\circ}{6} = 60^\circ$. The rotations that result in the figure coinciding with itself are $60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ, 360^\circ$. There are 6 such angles (including $360^\circ$).

Therefore, the order of rotational symmetry for a regular hexagon is 6.

The correct option is (D) 6.

Let's examine the symmetry properties of the uppercase letter 'Z' (in a standard print font).

Line Symmetry:

• Vertical line symmetry: If we draw a vertical line through the center of 'Z', the left and right halves are not mirror images.
• Horizontal line symmetry: If we draw a horizontal line through the center of 'Z', the top and bottom halves are not mirror images.

So, the letter 'Z' has no line symmetry.

Rotational Symmetry:

Let's consider rotating the letter 'Z' about its center. If we rotate 'Z' by $180^\circ$, the figure coincides exactly with its original position.

The smallest angle of rotation for which 'Z' looks the same is $180^\circ$.

The order of rotational symmetry is calculated as $\frac{360^\circ}{\text{smallest angle of rotation}}$.

Order $= \frac{360^\circ}{180^\circ} = 2$.

So, the letter 'Z' has rotational symmetry of order 2.

Based on this analysis:

• (A) Vertical line symmetry: False.
• (B) Horizontal line symmetry: False.
• (C) Rotational symmetry of order 2: True.
• (D) All of the above: False.

Therefore, the letter 'Z' has rotational symmetry of order 2.

The correct option is (C) Rotational symmetry of order 2.

Rotational symmetry of order greater than 1 means that the figure coincides with its original position after rotation by an angle $\theta$ where $0^\circ < \theta < 360^\circ$. If the order is 1, the only rotational symmetry is the $360^\circ$ rotation, which is considered to have no rotational symmetry in the conventional sense.

Let's examine the rotational symmetry for each shape:

(A) Isosceles trapezoid: An isosceles trapezoid has a pair of parallel sides and the non-parallel sides are equal in length. When rotated about its center, it only coincides with its original position after a $360^\circ$ rotation. Thus, its order of rotational symmetry is 1.

(B) Rectangle: A rectangle (unless it is a square) has rotational symmetry of order 2. It coincides with itself after a $180^\circ$ rotation about the intersection of its diagonals.

(C) Rhombus: A rhombus (unless it is a square) has rotational symmetry of order 2. It coincides with itself after a $180^\circ$ rotation about the intersection of its diagonals.

(D) Equilateral triangle: An equilateral triangle is a regular polygon with 3 sides. It has rotational symmetry of order 3, with the smallest angle of rotation being $120^\circ$.

We are looking for the shape that does NOT have rotational symmetry of order greater than 1.

• Isosceles trapezoid: Order = 1 (Not greater than 1)
• Rectangle: Order = 2 (Greater than 1)
• Rhombus: Order = 2 (Greater than 1)
• Equilateral triangle: Order = 3 (Greater than 1)

Therefore, the isosceles trapezoid is the shape that does not have rotational symmetry of order greater than 1.

The correct option is (A) Isosceles trapezoid.

Rotational symmetry means a figure coincides with itself after rotation about a fixed point by an angle $\theta$, where $0^\circ < \theta < 360^\circ$. The smallest such angle is called the angle of rotational symmetry, let's call it $\theta_{min}$.

The order of rotational symmetry ($n$) is the number of times the figure coincides with itself during a $360^\circ$ rotation. It is given by the formula:

$n = \frac{360^\circ}{\theta_{min}}$

The problem states that the figure coincides with its original position after a $180^\circ$ rotation. This means that $180^\circ$ is an angle of rotational symmetry. Therefore, the smallest angle of rotational symmetry, $\theta_{min}$, must be a divisor of $180^\circ$. This implies $\theta_{min} \leq 180^\circ$.

Since $\theta_{min} \leq 180^\circ$, and $\theta_{min} > 0^\circ$ (otherwise it would coincide for any small angle, like a circle), the order of rotational symmetry $n = \frac{360^\circ}{\theta_{min}}$ must satisfy:

$n = \frac{360^\circ}{\theta_{min}} \geq \frac{360^\circ}{180^\circ}$

$n \geq 2$

So, the order of rotational symmetry must be at least 2.

For example, if the smallest angle is $180^\circ$, the order is 2 (e.g., a rectangle or a parallelogram). If the smallest angle is $90^\circ$, the order is 4 (e.g., a square). In both cases, $180^\circ$ is also an angle of rotational symmetry ($1 \times 180^\circ$ and $2 \times 90^\circ = 180^\circ$).

If a figure has $180^\circ$ rotational symmetry, its order must be an even number ($2, 4, 6, \ldots$) and thus always greater than or equal to 2.

Therefore, if a figure coincides with its original position after a $180^\circ$ rotation, its order of rotational symmetry must be at least 2.

The correct option is (A) It must be at least 2.

Let's examine the symmetry of each letter mentioned in the options (assuming standard block letters):

A: Has vertical line symmetry.

B: Has horizontal line symmetry.

C: Has horizontal line symmetry.

D: Has horizontal line symmetry.

H: Has both horizontal and vertical line symmetry. It also has rotational symmetry of order 2.

I: Has both horizontal and vertical line symmetry. It also has rotational symmetry of order 2.

L: Has no line symmetry and rotational symmetry of order 1.

P: Has no line symmetry and rotational symmetry of order 1.

Now let's evaluate each option based on the symmetry type stated in the parenthesis:

(A) A and B (Vertical Line Symmetry): A has vertical line symmetry, but B does not. So, this statement is false.

(B) C and D (Horizontal Line Symmetry): C has horizontal line symmetry, and D also has horizontal line symmetry. So, this statement is true.

(C) H and I (Both Horizontal and Vertical Line Symmetry): H has both horizontal and vertical line symmetry, and I also has both horizontal and vertical line symmetry. So, this statement is true.

(D) L and P (Rotational Symmetry of Order 2): L has rotational symmetry of order 1, and P also has rotational symmetry of order 1. Neither has rotational symmetry of order 2. So, this statement is false.

Both option (B) and option (C) correctly identify a pair of letters where both letters possess the symmetry property described in the parenthesis. However, in a multiple-choice question with a single correct answer, option (C) describes the property of having a specific *combination* of symmetries, which is a distinct type of symmetry property ($D_2$ symmetry), whereas option (B) describes possessing a single type of line symmetry ($D_1$ symmetry with a horizontal axis).

Assuming the question intends to identify a pair sharing a specific type or combination of symmetry as described, both (B) and (C) are technically correct. However, the combination of both horizontal and vertical line symmetry is a notable property shared by H and I.

Based on common usage and the specific wording, option (C) is likely the intended answer as it highlights a specific combined symmetry property shared by the pair.

The correct option is (C) H and I (Both Horizontal and Vertical Line Symmetry).

A figure has horizontal line symmetry if there is a horizontal line such that folding the figure along this line divides it into two congruent mirror-image halves (the top half is a mirror image of the bottom half).

Let's examine the standard printed digits from 0 to 9:

• Digit 0: Has a horizontal line of symmetry.
• Digit 1: Does not have a horizontal line of symmetry (in standard print).
• Digit 2: Does not have a horizontal line of symmetry.
• Digit 3: Has a horizontal line of symmetry.
• Digit 4: Does not have a horizontal line of symmetry.
• Digit 5: Does not have a horizontal line of symmetry.
• Digit 6: Does not have a horizontal line of symmetry.
• Digit 7: Does not have a horizontal line of symmetry.
• Digit 8: Has a horizontal line of symmetry.
• Digit 9: Does not have a horizontal line of symmetry.

The digits that have horizontal line symmetry are 0, 3, and 8.

Comparing this list with the given options:

• (A) 0, 1, 8: Incorrect (1 does not have horizontal symmetry).
• (B) 0, 3, 8: Correct.
• (C) 0, 1, 3, 8: Incorrect (1 does not have horizontal symmetry).
• (D) 0, 1, 2, 5, 8: Incorrect (1, 2, and 5 do not have horizontal symmetry).

Therefore, the digits that have horizontal line symmetry are 0, 3, and 8.

The correct option is (B) 0, 3, 8.

A line of symmetry is a line that divides a figure into two congruent mirror-image halves.

A kite is a quadrilateral with two pairs of equal-length adjacent sides. In a standard kite (that is not a rhombus), the angles where the two pairs of equal sides meet are equal.

Consider the diagonals of a kite. One diagonal is the axis of symmetry. This diagonal connects the vertices between the unequal sides and is the perpendicular bisector of the other diagonal. Folding the kite along this main diagonal makes the equal adjacent sides and the equal opposite angles coincide, resulting in two congruent mirror halves.

The other diagonal of a kite is generally not a line of symmetry unless the kite is also a rhombus (where all four sides are equal, and both diagonals are lines of symmetry). Since the question specifies "not a rhombus", we are considering a kite where the sides are not all equal.

For a kite that is not a rhombus, only the main diagonal serves as a line of symmetry.

Therefore, a kite (not a rhombus) has exactly 1 line of symmetry.

The correct option is (B) 1.

The order of rotational symmetry ($n$) is the number of times a figure coincides with itself during a full $360^\circ$ rotation about its center.

If the smallest angle of rotational symmetry is $\theta_{min}$, the order is given by the formula:

$n = \frac{360^\circ}{\theta_{min}}$

In this question, the angle of rotational symmetry is given as $120^\circ$. We assume this is the smallest angle of rotation (if it weren't, then a smaller angle would be the angle of rotational symmetry, and $120^\circ$ would be a multiple of that smaller angle).

Using the formula with $\theta_{min} = 120^\circ$:

$n = \frac{360^\circ}{120^\circ}$

Calculation:

$\frac{360}{120} = 3$

So, the order of rotational symmetry is 3.

Therefore, if the angle of rotational symmetry is $120^\circ$, the order of rotational symmetry is 3.

The correct option is (C) 3.

We need to find the figure with the highest number of lines of symmetry among the given options.

Let's determine the number of lines of symmetry for each figure:

(A) Square: A square is a regular 4-sided polygon. It has 4 lines of symmetry.

(B) Equilateral Triangle: An equilateral triangle is a regular 3-sided polygon. It has 3 lines of symmetry.

(C) Rectangle: A rectangle (that is not a square) has 2 lines of symmetry. A square is a special type of rectangle, but since 'Square' is listed separately, 'Rectangle' here likely refers to non-square rectangles.

(D) Regular Octagon: A regular octagon is a regular 8-sided polygon. It has 8 lines of symmetry.

Comparing the number of lines of symmetry for each figure:

• Square: 4
• Equilateral Triangle: 3
• Rectangle: 2
• Regular Octagon: 8

The maximum number among these is 8.

Therefore, the regular octagon has the maximum number of lines of symmetry among the given options.

The correct option is (D) Regular Octagon.

Line symmetry, also known as reflectional symmetry, means that a figure can be folded along a line (the axis of symmetry) such that one half is the mirror image of the other half and they coincide perfectly.

Let the axis of symmetry be denoted by line $L$. Consider any point $P$ on the figure that is not on the line of symmetry. If the figure has line symmetry with respect to $L$, then there must be a corresponding point $P'$ on the figure such that $P'$ is the reflection of $P$ across the line $L$.

By the definition of reflection across a line, the line of reflection ($L$) is the perpendicular bisector of the segment joining the original point and its reflection. That is, the segment $PP'$ is perpendicular to the line $L$, and the line $L$ passes through the midpoint of the segment $PP'$.

This property holds for any pair of corresponding points in a figure with line symmetry, regardless of whether the figure is a polygon, a circle, or any other shape.

Therefore, if a figure has line symmetry, the axis of symmetry always passes through the midpoints of the segments joining corresponding points.

The statement is a fundamental property of line symmetry and is always true for any figure possessing this type of symmetry.

The correct option is (A) Always true.

We are looking for a figure that meets two specific criteria:

1. It has rotational symmetry of order 2.

2. It has exactly 2 lines of symmetry.

Let's analyze the symmetry properties of each option:

(A) Square: A square is a regular 4-sided polygon. It has rotational symmetry of order 4 (coincides after $90^\circ, 180^\circ, 270^\circ, 360^\circ$ rotations). It has 4 lines of symmetry (two diagonals and two lines joining midpoints of opposite sides).

(B) Rectangle (not a square): A rectangle that is not a square has unequal adjacent sides. It has rotational symmetry of order 2 (it coincides with its original position after a $180^\circ$ rotation about the intersection of its diagonals). It has exactly 2 lines of symmetry: the lines joining the midpoints of opposite sides.

(C) Equilateral triangle: An equilateral triangle is a regular 3-sided polygon. It has rotational symmetry of order 3 (coincides after $120^\circ, 240^\circ, 360^\circ$ rotations). It has 3 lines of symmetry (from each vertex to the midpoint of the opposite side).

(D) Regular pentagon: A regular pentagon is a regular 5-sided polygon. It has rotational symmetry of order 5 (coincides after $72^\circ, 144^\circ, 216^\circ, 288^\circ, 360^\circ$ rotations). It has 5 lines of symmetry (from each vertex to the midpoint of the opposite side).

Comparing the properties with the required conditions:

• Square: Order 4 (not 2), 4 lines (not 2). Does not fit.
• Rectangle (not a square): Order 2, exactly 2 lines. Fits the criteria.
• Equilateral triangle: Order 3 (not 2), 3 lines (not 2). Does not fit.
• Regular pentagon: Order 5 (not 2), 5 lines (not 2). Does not fit.

Only the rectangle (not a square) satisfies both conditions.

The correct option is (B) Rectangle (not a square).

Definition of Line Symmetry:

A figure is said to have line symmetry if it can be divided by a straight line into two parts such that one part is the mirror image of the other part. This dividing line is called the line of symmetry or the axis of symmetry.

Identifying Line Symmetry:

To identify if a figure has line symmetry, you can use the following method:

Imagine drawing a straight line through the figure. If you can fold the figure along this line such that the two halves match exactly, then the figure has line symmetry, and the line you drew is a line of symmetry.

Alternatively, you can imagine placing a mirror along the line. If the reflection of one side of the figure in the mirror perfectly matches the other side of the figure, then the figure has line symmetry.

A figure can have one or more lines of symmetry, or it may have no line symmetry at all.

A regular pentagon has 5 lines of symmetry.

In a regular polygon with an odd number of sides (like a pentagon which has 5 sides), each line of symmetry passes through a vertex and the midpoint of the side opposite that vertex.

For a regular pentagon, there are 5 vertices and 5 sides. Therefore, there are 5 lines of symmetry, each connecting one vertex to the midpoint of the opposite side.

Rough Sketch:

To show the lines of symmetry, draw a regular pentagon. Then, draw a line segment from each vertex to the midpoint of the side directly across from it. You will draw 5 such lines, and these are the lines of symmetry.

(Note: A visual sketch cannot be provided in this text format, but drawing lines from each vertex to the midpoint of the opposite side on a regular pentagon will show the 5 lines of symmetry).

A general parallelogram does not have line symmetry.

Explanation:

A figure has line symmetry if there exists a line such that if you fold the figure along this line, the two halves coincide perfectly. In a general parallelogram (where the adjacent sides are not equal and the angles are not $90^\circ$), there is no such line.

If you try to fold a parallelogram along a diagonal, the sides will not match up.

If you try to fold it along a line passing through the midpoints of opposite sides, the vertices will not coincide unless it is a rectangle (in which case the angles are $90^\circ$) or a rhombus (in which case all sides are equal).

Therefore, unless a parallelogram is a special type like a rectangle or a rhombus, it does not possess line symmetry.

Special Cases:

• A rectangle has 2 lines of symmetry (lines joining the midpoints of opposite sides).
• A rhombus has 2 lines of symmetry (the diagonals).
• A square (which is both a rectangle and a rhombus) has 4 lines of symmetry (the diagonals and the lines joining the midpoints of opposite sides).

However, for a general parallelogram, the number of lines of symmetry is zero.

An isosceles trapezoid (non-parallelogram) has 1 line of symmetry.

Explanation:

An isosceles trapezoid is a trapezoid where the non-parallel sides (legs) are equal in length. This equality of legs leads to one line of symmetry.

The line of symmetry for an isosceles trapezoid is the line segment that connects the midpoint of the longer base to the midpoint of the shorter base.

If you fold the isosceles trapezoid along this line, the two non-parallel sides will coincide, and the segments of the bases on either side of the line will also coincide due to the equal base angles.

Rough Sketch:

Draw a trapezoid with one pair of parallel sides (top and bottom). Make the non-parallel sides equal in length. Find the midpoint of the top side and the midpoint of the bottom side. Draw a vertical line segment connecting these two midpoints. This line is the single line of symmetry.

(Note: A visual sketch cannot be provided in this text format, but drawing a line connecting the midpoints of the two parallel bases on an isosceles trapezoid will show the single line of symmetry).

The digits that have exactly one line of symmetry are 1 and 3.

Digit 1:

The digit 1 has one line of symmetry, which is a vertical line passing through the center of the digit.

Imagine drawing a vertical line down the middle of the digit '1'. If you fold it along this line, the left side would match the right side.

Digit 3:

The digit 3 has one line of symmetry, which is a horizontal line passing through the center of the digit.

Imagine drawing a horizontal line across the middle of the digit '3'. If you fold it along this line, the top half would match the bottom half.

The other digits have either more than one line of symmetry (like 0 and 8) or no line of symmetry (like 2, 4, 5, 6, 7, 9).

Definition of Rotational Symmetry:

A figure is said to have rotational symmetry if, after being rotated by some angle less than $360^\circ$ about a fixed point, it looks exactly the same as the original figure. The fixed point is called the centre of rotation, and the angle of rotation is called the angle of rotational symmetry.

Key Components of Rotational Symmetry:

The two key components required to describe rotational symmetry are:

1. Centre of Rotation: This is the fixed point around which the figure is rotated. For many geometric shapes, this point is the geometric centre of the figure.

2. Order of Rotational Symmetry (or Angle of Rotational Symmetry): This describes how many times the figure coincides with itself during a full $360^\circ$ rotation. If a figure coincides with itself $n$ times (where $n > 1$) during a $360^\circ$ rotation, its order of rotational symmetry is $n$. The angle of rotational symmetry is the smallest angle through which the figure must be rotated to coincide with itself, calculated as $\frac{360^\circ}{\text{Order}}$.

For a square:

Centre of Rotation:

The centre of rotation for a square is the point where its two diagonals intersect. This point is the geometric centre of the square.

Angle of Rotation:

The angle of rotation is the smallest angle by which the square can be rotated about its centre to coincide with its original position.

A square has 4 equal sides and 4 equal $90^\circ$ angles. It has rotational symmetry of order 4.

This means it coincides with itself after rotations of $90^\circ, 180^\circ, 270^\circ$, and $360^\circ$.

The smallest non-zero angle of rotation is $90^\circ$.

The angle of rotation = $\frac{360^\circ}{\text{Order of rotational symmetry}}$

Angle of rotation = $\frac{360^\circ}{4} = 90^\circ$.

So, the angle of rotation for a square is $90^\circ$.

Definition of Order of Rotational Symmetry:

The order of rotational symmetry is the number of times a figure coincides with itself during a complete rotation of $360^\circ$ about its centre of rotation, including the original position.

An order of rotational symmetry of 1 means the figure only looks the same after a full $360^\circ$ rotation, which indicates no rotational symmetry (except for the trivial case).

Order of Rotational Symmetry for a Square:

Consider a square rotated about its centre (the intersection of its diagonals).

The square will coincide with its original position after rotating by:

• $90^\circ$
• $180^\circ$
• $270^\circ$
• $360^\circ$ (back to the original position)

It looks the same at these four distinct angles of rotation within a $360^\circ$ turn.

Therefore, the order of rotational symmetry for a square is 4.

Yes, a circle does have rotational symmetry.

Explanation:

A circle is perfectly symmetrical around its centre. If you rotate a circle about its centre by any angle, it will look exactly the same as the original circle.

Centre of Rotation:

The centre of rotation for a circle is its centre.

Order of Rotational Symmetry:

Since a circle coincides with itself after rotation by any angle about its centre, it matches itself infinitely many times during a $360^\circ$ rotation. Therefore, the order of rotational symmetry for a circle is infinite.

Angle of Rotation:

Because the circle coincides with itself after rotation by any angle (other than $0^\circ$ which is the original position), there isn't a single smallest angle of rotation. It has rotational symmetry for every angle of rotation about its centre.

For an equilateral triangle:

Order of Rotational Symmetry:

An equilateral triangle is a regular polygon with 3 equal sides and 3 equal angles. It has rotational symmetry about its geometric centre.

If you rotate an equilateral triangle about its centre, it will coincide with its original position after rotations by $\frac{360^\circ}{3} = 120^\circ$, $2 \times 120^\circ = 240^\circ$, and $3 \times 120^\circ = 360^\circ$ (which is the original position).

It matches itself at 3 distinct positions during a $360^\circ$ rotation.

Therefore, the order of rotational symmetry for an equilateral triangle is 3.

Angle of Rotation:

The angle of rotation is the smallest angle by which the figure can be rotated about its centre to coincide with its original position.

Angle of rotation = $\frac{360^\circ}{\text{Order of rotational symmetry}}$

Angle of rotation = $\frac{360^\circ}{3} = 120^\circ$.

So, the angle of rotation for an equilateral triangle is $120^\circ$.

No, a scalene triangle does not have rotational symmetry (except for the trivial rotation of $360^\circ$).

Explanation:

A scalene triangle is a triangle in which all three sides are of different lengths, and consequently, all three angles are of different measures.

For a figure to have rotational symmetry (of order greater than 1), it must look exactly the same after being rotated by an angle less than $360^\circ$ about its centre.

Due to the unequal sides and angles of a scalene triangle, there is no angle of rotation (other than $0^\circ$ or multiples of $360^\circ$) for which the triangle will coincide with its original position when rotated about its geometric centre.

The only rotation that maps a scalene triangle onto itself is a rotation of $360^\circ$, which brings the figure back to its starting position. This corresponds to an order of rotational symmetry of 1, which is considered trivial and means the figure does not possess non-trivial rotational symmetry.

Therefore, a scalene triangle has no rotational symmetry.

The capital English letters that have rotational symmetry of order greater than 1 are:

• H (Order 2)
• I (Order 2)
• N (Order 2)
• O (Infinite Order)
• S (Order 2)
• X (Order 2)
• Z (Order 2)

These letters look the same after being rotated by a certain angle (less than $360^\circ$) about their centre.

For letters H, I, N, S, X, and Z, the angle of rotation is $180^\circ$, meaning they coincide with their original position twice in a $360^\circ$ rotation (at $180^\circ$ and $360^\circ$), giving an order of 2.

The letter O has rotational symmetry for any angle of rotation about its centre, similar to a circle, hence its order is infinite.

The angle of rotation for a figure with rotational symmetry of order $n$ is given by the formula:

Angle of rotation = $\frac{360^\circ}{\text{Order of rotational symmetry}}$

In this case, the order of rotational symmetry is given as 4.

So, the angle of rotation = $\frac{360^\circ}{4}$.

Calculating the value:

$360 \div 4 = 90$

Therefore, the angle of rotation is $90^\circ$.

This means the figure coincides with its original position after being rotated by $90^\circ$ about its centre.

The relationship between the angle of rotation and the order of rotational symmetry is given by:

Angle of rotation = $\frac{360^\circ}{\text{Order of rotational symmetry}}$

We are given that the angle of rotation is $90^\circ$. Let the order of rotational symmetry be $n$.

So, we have the equation:

$90^\circ = \frac{360^\circ}{n}$

To find $n$, we can rearrange the equation:

$n = \frac{360^\circ}{90^\circ}$

$n = \frac{360}{90}$

$n = 4$

Therefore, the order of rotational symmetry for the figure is 4.

A real-life example of an object that has line symmetry but not rotational symmetry of order greater than 1 is a common arrowhead.

Line Symmetry:

An arrowhead typically has one line of symmetry. This line runs vertically down the center of the arrowhead, from the tip to the base. If you folded the arrowhead along this line, the two sides would match perfectly, making it a mirror image.

Rotational Symmetry:

If you rotate a common arrowhead about its center by any angle less than $360^\circ$ (other than $0^\circ$), it will not coincide with its original position. For instance, rotating it by $180^\circ$ would make it point downwards, which is clearly different from pointing upwards.

It only looks the same again after a full $360^\circ$ rotation.

Therefore, its order of rotational symmetry is 1, which means it does not have rotational symmetry of order greater than 1.

Other examples include:

• A spoon (seen from directly above)
• The capital letter 'A'
• An isosceles trapezoid (non-parallelogram)

A real-life example of an object that has both line symmetry and rotational symmetry of order greater than 1 is a square.

Examples of square objects in real life include:

• A standard chessboard
• Many window panes
• Some types of tiles

Line Symmetry:

A square has 4 lines of symmetry:

• Two lines passing through the midpoints of opposite sides.
• Two lines passing through opposite vertices (the diagonals).

Folding a square along any of these lines makes the two halves coincide perfectly.

Rotational Symmetry:

A square has rotational symmetry about its center (the intersection of its diagonals). It coincides with its original position after rotations of $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$.

The smallest angle of rotation is $90^\circ$.

The order of rotational symmetry is the number of times it coincides in a $360^\circ$ rotation, which is 4.

Order = $\frac{360^\circ}{90^\circ} = 4$.

Since the order (4) is greater than 1, a square has rotational symmetry of order greater than 1.

Thus, a square is an object that possesses both line symmetry and rotational symmetry of order greater than 1.

Other examples include:

• A regular hexagon (6 lines of symmetry, order 6)
• A starfish (typically 5 lines of symmetry, order 5)
• A circle (infinite lines of symmetry, infinite order)

A regular hexagon is a polygon with 6 equal sides and 6 equal interior angles. It has rotational symmetry about its centre.

The order of rotational symmetry for a regular polygon with $n$ sides is $n$. For a regular hexagon, $n=6$.

So, the order of rotational symmetry for a regular hexagon is 6.

The smallest angle through which a figure can be rotated to look the same (the angle of rotational symmetry) is given by the formula:

Angle of rotation = $\frac{360^\circ}{\text{Order of rotational symmetry}}$

Substituting the order of rotational symmetry (6):

Angle of rotation = $\frac{360^\circ}{6}$

Calculating the value:

$360 \div 6 = 60$

Therefore, the smallest angle through which a regular hexagon can be rotated about its centre to look the same is $60^\circ$.

A rhombus is a quadrilateral with all four sides of equal length. Its opposite angles are equal, and its diagonals bisect each other at right angles.

Lines of Symmetry:

A rhombus has exactly 2 lines of symmetry.

These lines of symmetry are the two diagonals of the rhombus.

If you fold a rhombus along either of its diagonals, the two halves will coincide perfectly.

Rotational Symmetry:

Yes, a rhombus does have rotational symmetry.

The centre of rotation is the point where the two diagonals intersect.

When you rotate a rhombus about its centre by $180^\circ$, it will look exactly the same as the original rhombus. It also looks the same after a $360^\circ$ rotation.

Order of Rotational Symmetry:

A rhombus coincides with its original position twice during a $360^\circ$ rotation: at $180^\circ$ and at $360^\circ$.

Therefore, the order of rotational symmetry for a rhombus is 2.

The angle of rotation is $\frac{360^\circ}{\text{Order}} = \frac{360^\circ}{2} = 180^\circ$.

In summary, a rhombus has 2 lines of symmetry and rotational symmetry of order 2.

No, a kite (that is not a rhombus) does not have rotational symmetry of order greater than 1.

Explanation:

A kite is a quadrilateral with two distinct pairs of equal adjacent sides. In a kite that is not a rhombus, the lengths of the two pairs of adjacent sides are different.

A kite has one line of symmetry (the diagonal connecting the vertices where the equal sides meet). The diagonals are perpendicular, and one diagonal is bisected by the other.

The potential centre of rotation would be the intersection of the diagonals. However, if you rotate a kite (not a rhombus) about this point by any angle less than $360^\circ$ (other than $0^\circ$), the figure will not coincide with its original position.

For example, a $180^\circ$ rotation would swap opposite vertices. In a kite, the opposite angles are not all equal (only one pair of opposite angles between unequal sides are equal). Swapping vertices will result in a figure that does not match the original kite unless all sides are equal (making it a rhombus) or all angles are equal (making it a square), which are special cases of a kite/rhombus/rectangle.

Since a kite (not a rhombus) only coincides with its original position after a $360^\circ$ rotation, its order of rotational symmetry is 1. An order of 1 means it lacks rotational symmetry of order greater than 1 (non-trivial rotational symmetry).

Thus, a kite (not a rhombus) does not have rotational symmetry of order greater than 1.

Yes, a figure can have rotational symmetry of order greater than 1 but no line symmetry.

Example:

The capital English letter N is a common example of a figure that has rotational symmetry but no line symmetry.

Explanation:

Rotational Symmetry:

The letter 'N' has rotational symmetry of order 2 about its centre. If you rotate the letter 'N' by $180^\circ$ about its centre, it will look exactly the same as the original letter.

Order of rotational symmetry = $\frac{360^\circ}{\text{Angle of rotation}} = \frac{360^\circ}{180^\circ} = 2$.

Since the order (2) is greater than 1, it has rotational symmetry.

Line Symmetry:

If you try to draw a line through the letter 'N' such that folding along this line results in two mirror images, you will find no such line. A vertical line would make the left and right halves different. A horizontal line would make the top and bottom halves different. There is no line that can divide the 'N' into two mirror-image halves.

Therefore, the letter 'N' has no line symmetry.

Other examples include the letters S and Z, and the swastika symbol.

A regular octagon is a polygon with 8 equal sides and 8 equal interior angles.

Lines of Symmetry:

For any regular polygon with $n$ sides, the number of lines of symmetry is equal to the number of sides, $n$.

A regular octagon has $n=8$ sides.

Therefore, a regular octagon has 8 lines of symmetry.

These lines pass either through opposite vertices or through the midpoints of opposite sides.

Order of Rotational Symmetry:

For any regular polygon with $n$ sides, the order of rotational symmetry is also equal to the number of sides, $n$.

A regular octagon has $n=8$ sides.

Therefore, the order of rotational symmetry for a regular octagon is 8.

This means it coincides with its original position 8 times during a full $360^\circ$ rotation about its centre.

The angle of rotation is $\frac{360^\circ}{8} = 45^\circ$.

A fan with 3 blades arranged equally around the centre has rotational symmetry.

Angle of Rotation:

Since the 3 blades are arranged equally around the full circle ($360^\circ$), the angle between successive positions where the fan looks the same is the total angle divided by the number of blades.

Angle of rotation = $\frac{360^\circ}{\text{Number of blades}}$

Angle of rotation = $\frac{360^\circ}{3}$

Calculating the value:

$360 \div 3 = 120$

The angle of rotation is $120^\circ$. This is the smallest angle through which the fan must be rotated to look the same as its original position.

Order of Rotational Symmetry:

The order of rotational symmetry is the number of times the figure coincides with itself during a full $360^\circ$ rotation.

Since the angle of rotation is $120^\circ$, the fan will look the same after rotations of $120^\circ$, $240^\circ$, and $360^\circ$ (which is the original position).

It coincides with itself 3 times.

Order of rotational symmetry = $\frac{360^\circ}{\text{Angle of rotation}}$

Order of rotational symmetry = $\frac{360^\circ}{120^\circ}$

Order of rotational symmetry = 3

The order of rotational symmetry is 3.

Concept of Line Symmetry:

Line symmetry, also known as reflectional symmetry, occurs when a figure can be folded along a straight line such that one half is the mirror image of the other half. This straight line is called the line of symmetry or axis of symmetry.

If you were to place a mirror along the line of symmetry, the reflection of one side would perfectly overlap the other side of the figure.

Lines of Symmetry for Specific Shapes:

1. Square:

A square is a quadrilateral with four equal sides and four right angles ($90^\circ$).

It has 4 lines of symmetry.

• Two lines of symmetry pass through the midpoints of opposite sides (one vertical, one horizontal).
• Two lines of symmetry pass through opposite vertices (the two diagonals).

(Imagine a square; draw a line straight down the middle, one straight across the middle, and one along each diagonal. These are the 4 lines).

2. Rectangle (not a square):

A rectangle is a quadrilateral with four right angles ($90^\circ$) and opposite sides equal. In a rectangle that is not a square, adjacent sides are unequal.

It has 2 lines of symmetry.

• One line of symmetry passes through the midpoints of the longer pair of opposite sides.
• One line of symmetry passes through the midpoints of the shorter pair of opposite sides.

(Imagine a rectangle; draw a line straight down the middle and one straight across the middle. The diagonals are not lines of symmetry for a non-square rectangle).

3. Rhombus (not a square):

A rhombus is a quadrilateral with four equal sides and opposite angles equal. In a rhombus that is not a square, the angles are not $90^\circ$.

It has 2 lines of symmetry.

• The two lines of symmetry are the two diagonals of the rhombus.

(Imagine a rhombus; draw lines along each of its diagonals. These are the 2 lines of symmetry. The lines through the midpoints of opposite sides are not lines of symmetry for a non-square rhombus).

Concept of Rotational Symmetry:

Rotational symmetry is a type of symmetry where a figure looks the same after a certain amount of rotation about a fixed point. Unlike line symmetry, where the figure is reflected across a line, rotational symmetry involves turning the figure.

A figure has rotational symmetry if, when rotated by an angle less than $360^\circ$ around a central point, it perfectly coincides with its original position.

Components of Rotational Symmetry:

1. Centre of Rotation: This is the fixed point around which the figure is rotated. This point may be inside or outside the figure. For many standard shapes like polygons and circles, the centre of rotation is the geometric centre of the figure.

2. Angle of Rotation: This is the smallest angle, less than $360^\circ$, through which the figure must be rotated about the centre of rotation so that it appears exactly the same as it did originally. The angle of rotation is sometimes called the least angle of rotational symmetry.

3. Order of Rotational Symmetry: This is the number of times a figure coincides with its original position during a full $360^\circ$ rotation about its centre of rotation. It includes the original position itself as one count. The order of rotational symmetry ($n$) and the angle of rotation ($\theta$) are related by the formula: $n = \frac{360^\circ}{\theta}$. The order must be an integer greater than 1 for the figure to have non-trivial rotational symmetry.

Example: Rotational Symmetry of a Rectangle (not a square):

Consider a rectangle ABCD.

Centre of Rotation:

The centre of rotation for a rectangle is the point where its two diagonals intersect. Let's call this point O.

Angle of Rotation and Order:

Rotate the rectangle ABCD clockwise about its centre O.

• After a $90^\circ$ rotation, vertex A would move to where B was, B to C, C to D, and D to A. A rectangle does not look the same after a $90^\circ$ rotation (unless it's a square).
• After a $180^\circ$ rotation, vertex A moves to where C was, B to D, C to A, and D to B. The rectangle looks exactly the same as the original one.
• After a $270^\circ$ rotation, it will not look the same (similar to the $90^\circ$ rotation).
• After a $360^\circ$ rotation, it returns to its original position.

The smallest non-zero angle of rotation that makes the rectangle coincide with itself is $180^\circ$.

Therefore, the angle of rotation is $180^\circ$.

The figure coincides with itself 2 times during a $360^\circ$ rotation (at $180^\circ$ and $360^\circ$).

The order of rotational symmetry is $\frac{360^\circ}{180^\circ} = 2$.

The order of rotational symmetry for a rectangle is 2.

Positions After Rotation:

Let the vertices be labeled A, B, C, D in a clockwise direction starting from the top-left.

• Original position: Vertices are at positions corresponding to A, B, C, D.
• After $180^\circ$ rotation: Vertex A is now at the original position of C, B is at D, C is at A, and D is at B. The rectangle appears identical to the original.
• After $360^\circ$ rotation: Vertices are back at their original positions A, B, C, D. The rectangle is identical.

The rectangle coincides with itself at $180^\circ$ and $360^\circ$ rotations.

An equilateral triangle is a triangle with all three sides of equal length and all three interior angles equal to $60^\circ$.

Lines of Symmetry:

An equilateral triangle has 3 lines of symmetry.

Each line of symmetry passes through a vertex and the midpoint of the opposite side. These lines are also the altitudes, medians, and angle bisectors of the equilateral triangle.

(Imagine an equilateral triangle; draw a line from each corner straight down to the middle of the opposite side. There are three such lines, and they intersect at the centre of the triangle).

Rotational Symmetry:

An equilateral triangle has rotational symmetry about its geometric centre (the point where the lines of symmetry intersect).

The smallest angle of rotation required to make the triangle look the same is the angle obtained by dividing the total angle of a circle ($360^\circ$) by the number of vertices (or sides, which is 3 for an equilateral triangle).

Angle of rotation = $\frac{360^\circ}{3} = 120^\circ$.

When rotated by $120^\circ$, one vertex moves to the position of the next vertex, and the triangle coincides with its original appearance.

The triangle will also look the same after rotations of $2 \times 120^\circ = 240^\circ$ and $3 \times 120^\circ = 360^\circ$.

Order of Rotational Symmetry:

The order of rotational symmetry is the number of times the figure coincides with its original position during a $360^\circ$ rotation.

For an equilateral triangle, this happens at $120^\circ$, $240^\circ$, and $360^\circ$.

So, the figure coincides with itself 3 times.

The order of rotational symmetry is 3.

Order = $\frac{360^\circ}{\text{Angle of rotation}} = \frac{360^\circ}{120^\circ} = 3$.

Let's analyze the capital English letters H, S, and Z.

Letter H:

(a) Line Symmetry: Yes, the letter H has line symmetry.

• It has a vertical line of symmetry passing through its centre. (Folding along this line makes the left half match the right half).
• It has a horizontal line of symmetry passing through its centre. (Folding along this line makes the top half match the bottom half).

Number of lines of symmetry = 2.

(Imagine the letter H; draw a line down the middle and a line across the middle).

(b) Rotational Symmetry: Yes, the letter H has rotational symmetry of order greater than 1.

When rotated by $180^\circ$ about its centre, the letter H looks the same.

Order of rotational symmetry = $\frac{360^\circ}{180^\circ} = 2$.

Angle of rotation = $180^\circ$.

Letter S:

(a) Line Symmetry: No, the letter S does not have line symmetry.

If you try to draw a vertical or horizontal line through it, folding along that line will not make the two halves match (unless the letter is drawn in a highly stylized way that enforces symmetry).

(b) Rotational Symmetry: Yes, the letter S has rotational symmetry of order greater than 1.

When rotated by $180^\circ$ about its centre, the letter S looks the same.

Order of rotational symmetry = $\frac{360^\circ}{180^\circ} = 2$.

Angle of rotation = $180^\circ$.

Letter Z:

(a) Line Symmetry: No, the letter Z does not have line symmetry.

Similar to 'S', there is no line along which 'Z' can be folded to produce two mirror images.

(b) Rotational Symmetry: Yes, the letter Z has rotational symmetry of order greater than 1.

When rotated by $180^\circ$ about its centre, the letter Z looks the same.

Order of rotational symmetry = $\frac{360^\circ}{180^\circ} = 2$.

Angle of rotation = $180^\circ$.

A regular hexagon is a polygon with 6 equal sides and 6 equal interior angles. It is both equilateral and equiangular.

Lines of Symmetry:

A regular hexagon has 6 lines of symmetry.

These lines are of two types:

• Lines that pass through opposite vertices. There are 3 such lines.
• Lines that pass through the midpoints of opposite sides. There are 3 such lines.

(Imagine a regular hexagon; draw lines connecting opposite corners and lines connecting the midpoints of opposite edges. You will draw a total of 6 lines).

Rotational Symmetry:

A regular hexagon has rotational symmetry about its geometric centre (the point where all the lines of symmetry intersect).

The smallest angle of rotation required to make the hexagon coincide with its original position is:

Angle of rotation = $\frac{360^\circ}{\text{Number of sides}}$

Angle of rotation = $\frac{360^\circ}{6} = 60^\circ$.

When rotated by $60^\circ$ about its centre, each vertex moves to the position of the next vertex, and the hexagon looks identical.

The hexagon will coincide with its original position after rotations of $60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ$, and $360^\circ$.

Order of Rotational Symmetry:

The order of rotational symmetry is the number of times the figure coincides with its original position during a full $360^\circ$ rotation.

For a regular hexagon, this happens 6 times.

The order of rotational symmetry is 6.

Order = $\frac{360^\circ}{\text{Angle of rotation}} = \frac{360^\circ}{60^\circ} = 6$.

Relationship for Regular Polygons:

For any regular polygon with $n$ sides, there is a direct and simple relationship between the number of lines of symmetry and the order of rotational symmetry.

The number of lines of symmetry of a regular polygon with $n$ sides is exactly equal to $n$.

The order of rotational symmetry of a regular polygon with $n$ sides is also exactly equal to $n$.

Thus, for regular polygons, the number of lines of symmetry is equal to the order of rotational symmetry, and both are equal to the number of sides of the polygon.

Number of lines of symmetry = Order of rotational symmetry = Number of sides ($n$).

Illustration with Examples:

Example 1: Square

A square is a regular polygon with 4 sides. So, $n=4$.

• Number of lines of symmetry: A square has 4 lines of symmetry (2 through opposite vertices, 2 through midpoints of opposite sides). This matches $n=4$.
• Order of rotational symmetry: A square has rotational symmetry of order 4 (rotates onto itself at $90^\circ, 180^\circ, 270^\circ, 360^\circ$). This matches $n=4$.

Here, Number of lines of symmetry (4) = Order of rotational symmetry (4) = Number of sides (4).

Example 2: Regular Pentagon

A regular pentagon is a regular polygon with 5 sides. So, $n=5$.

• Number of lines of symmetry: A regular pentagon has 5 lines of symmetry (each connecting a vertex to the midpoint of the opposite side). This matches $n=5$.
• Order of rotational symmetry: A regular pentagon has rotational symmetry of order 5 (rotates onto itself at $\frac{360^\circ}{5} = 72^\circ$ and multiples). This matches $n=5$.

Here, Number of lines of symmetry (5) = Order of rotational symmetry (5) = Number of sides (5).

This relationship holds true for all regular polygons (e.g., equilateral triangle ($n=3$), regular hexagon ($n=6$), regular octagon ($n=8$), etc.).

Real-Life Objects/Designs with Line Symmetry:

These objects can be divided into two mirror-image halves by a line.

(Note: Actual drawings cannot be provided in this format, but you can visualize or sketch them).

• A Butterfly: A butterfly typically has one vertical line of symmetry down its body, through the centre of its wings.
• A Human Face: A human face, ideally, has one vertical line of symmetry down the middle (though perfect symmetry is rare in reality).
• A Chair: Many common chair designs have a vertical line of symmetry down the centre.

Real-Life Objects/Designs with Rotational Symmetry (Order > 1):

These objects look the same after being rotated by an angle less than $360^\circ$ about a central point.

(Note: Actual drawings cannot be provided in this format, but you can visualize or sketch them).

• A Pinwheel (with equally spaced blades): A pinwheel with, say, 4 identical blades equally spaced has rotational symmetry of order 4 ($90^\circ$ rotation).
• A Steering Wheel: A steering wheel with spokes equally spaced around the centre has rotational symmetry. The order depends on the number of equally spaced spokes (e.g., a 3-spoke wheel has order 3, a 4-spoke wheel has order 4).
• A Ceiling Fan (with identical blades): A ceiling fan with, say, 3 identical blades equally spaced has rotational symmetry of order 3 ($120^\circ$ rotation).

A figure that has rotational symmetry of order 2 but no line symmetry is the capital English letter S (as discussed previously), or the letter Z, or the letter N.

Another geometric example is a parallelogram that is not a rectangle or a rhombus.

Let's use the letter S as the example.

(Note: A visual representation is needed here. Imagine the letter 'S').

Rotational Symmetry (Order 2):

The letter 'S' has a geometric centre. If you rotate the letter 'S' by $180^\circ$ around this centre, the figure will land exactly on itself. The top curve will move to the position of the bottom curve, and vice versa, but the overall shape and orientation will be the same.

The smallest angle of rotation for which it looks the same is $180^\circ$.

Order of rotational symmetry = $\frac{360^\circ}{180^\circ} = 2$.

Since the order (2) is greater than 1, it has rotational symmetry of order 2.

Lack of Line Symmetry:

To have line symmetry, there must be a line that divides the figure into two mirror images.

For the letter 'S', if you draw a vertical line through its centre, the left side is not the mirror image of the right side.

If you draw a horizontal line through its centre, the top half is not the mirror image of the bottom half.

There is no line that would create a perfect mirror reflection across it for the shape of the letter 'S'.

Therefore, the letter 'S' has no line symmetry.

A scalene triangle is defined as a triangle in which all three sides have different lengths, and consequently, all three interior angles have different measures.

Lack of Line Symmetry:

For a figure to have line symmetry, there must be at least one line that acts as a mirror, dividing the figure into two identical halves.

In a scalene triangle, since all sides are of different lengths and all angles are of different measures, there is no line that can be drawn to divide the triangle into two congruent (identical) halves that are mirror images of each other. Any attempt to fold a scalene triangle along any line will result in the two parts not coinciding perfectly.

Therefore, a scalene triangle has no line symmetry.

Lack of Rotational Symmetry of Order Greater than 1:

Rotational symmetry of order greater than 1 exists if a figure can be rotated about its centre by an angle less than $360^\circ$ and still appear exactly the same as the original figure.

The potential centre of rotation for a triangle is typically its centroid (the intersection of its medians). If you rotate a scalene triangle about its centroid, because all sides and angles are unequal, it will not coincide with its original position for any rotation angle between $0^\circ$ and $360^\circ$.

The only rotation that maps a scalene triangle onto itself is a full $360^\circ$ rotation, which brings it back to its starting position. This corresponds to an order of rotational symmetry of 1 (trivial symmetry), meaning it does not have rotational symmetry of order greater than 1.

Therefore, a scalene triangle has no rotational symmetry (of order greater than 1).

Let's analyze the digits 0 and 8.

Digit 0:

Lines of Symmetry:

The digit 0 (often drawn as an oval or circle) has two lines of symmetry.

• A vertical line of symmetry passing through the centre.
• A horizontal line of symmetry passing through the centre.

If you fold the digit '0' along either of these lines, the two halves are mirror images and coincide perfectly.

Rotational Symmetry:

The digit 0 has rotational symmetry about its centre. Similar to a circle, if you rotate '0' about its centre, it will look the same at infinitely many angles.

The order of rotational symmetry for the digit 0 is infinite.

It looks the same after rotation by any angle about its centre.

Digit 8:

Lines of Symmetry:

The digit 8 has two lines of symmetry.

• A vertical line of symmetry passing through the centre.
• A horizontal line of symmetry passing through the centre.

If you fold the digit '8' along either of these lines, the two loops are divided symmetrically, and the two halves coincide.

Rotational Symmetry:

The digit 8 has rotational symmetry about its centre (the point where the two loops meet in the middle).

If you rotate the digit '8' by $180^\circ$ about its centre, it will look exactly the same as the original digit. The top loop will swap positions with the bottom loop, but the figure remains unchanged.

The smallest angle of rotation is $180^\circ$.

Order of rotational symmetry = $\frac{360^\circ}{\text{Angle of rotation}} = \frac{360^\circ}{180^\circ} = 2$.

The order of rotational symmetry for the digit 8 is 2.

Consider a windmill with 4 identical blades arranged equally around a central hub.

(Note: A diagram cannot be provided in this text format. Imagine a cross shape (+) with blades at the ends).

Line Symmetry:

A windmill with 4 identical, equally spaced blades has 4 lines of symmetry.

These lines are:

• Two lines that pass through the tips of opposite blades.
• Two lines that pass between adjacent blades.

If you fold the windmill design along any of these four lines, the two halves will be mirror images and will coincide perfectly.

Rotational Symmetry:

A windmill with identical, equally spaced blades exhibits rotational symmetry.

Centre of Rotation: The centre of rotation is the central point where the blades are attached (the axle or hub of the windmill).

Angle of Rotation: Since there are 4 equally spaced blades around $360^\circ$, the angle between successive positions where the windmill looks the same is:

Angle of rotation = $\frac{360^\circ}{\text{Number of blades}}$

Angle of rotation = $\frac{360^\circ}{4} = 90^\circ$.

The smallest angle through which the windmill can be rotated to look the same is $90^\circ$.

Order of Rotational Symmetry: The order of rotational symmetry is the number of times the windmill coincides with its original position during a $360^\circ$ rotation.

Order of rotational symmetry = $\frac{360^\circ}{\text{Angle of rotation}}$

Order of rotational symmetry = $\frac{360^\circ}{90^\circ} = 4$.

The windmill looks the same after rotations of $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$. Thus, the order is 4.

In summary, a windmill with 4 identical, equally spaced blades has 4 lines of symmetry and rotational symmetry of order 4 with an angle of rotation of $90^\circ$ about its central hub.

Design Description:

Let's create a design consisting of an equilateral triangle with a circle inscribed perfectly inside it (tangent to all three sides), centered at the centroid of the triangle.

(Note: A visual representation is needed here. Imagine an equilateral triangle with a circle drawn inside it, touching each side at its midpoint).

Explanation of Symmetries:

Line Symmetry:

This design has 3 lines of symmetry.

Each line of symmetry passes through a vertex of the equilateral triangle and the midpoint of the opposite side. These lines also pass through the center of the inscribed circle. If you fold the figure along any of these three lines, the two halves of both the triangle and the circle will coincide perfectly, creating a mirror image.

Since 3 is at least 2, the condition of having at least two lines of symmetry is met.

Rotational Symmetry:

This design has rotational symmetry about the centre of the circle (which is also the centroid of the equilateral triangle).

Since the outer shape is an equilateral triangle and the inner shape (circle) is perfectly centered, the entire design will look the same after certain rotations.

The smallest angle of rotation that makes the equilateral triangle coincide with its original position is $120^\circ$. The circle, being perfectly round and centered, looks the same after any rotation. Thus, the combination looks the same after a $120^\circ$ rotation.

The angle of rotation is $120^\circ$.

The order of rotational symmetry is the number of times the design coincides with its original position during a $360^\circ$ rotation.

Order of rotational symmetry = $\frac{360^\circ}{\text{Angle of rotation}} = \frac{360^\circ}{120^\circ} = 3$.

Since the order (3) is at least 3, the condition for rotational symmetry is met.

Therefore, the design of an equilateral triangle with an inscribed circle has 3 lines of symmetry and rotational symmetry of order 3.