Chapter 14 Practical Geometry (Additional Questions)

The correct option is (D).

The geometrical instrument specifically designed and used for drawing circles and arcs is a Compass.

A compass has two arms, one with a sharp point (pivot) and the other with a pencil or pen attachment. By fixing the pivot at the desired center of the circle and rotating the arm with the pencil while keeping the distance between the arms constant, a circle of a specific radius can be drawn.

Let's briefly look at the other options:

(A) Ruler: A ruler is used for drawing straight lines and measuring lengths.

(B) Protractor: A protractor is used for measuring and drawing angles.

(C) Divider: A divider is used for transferring measurements or dividing a line segment into equal parts. It has two sharp points and is not used for drawing curves.

Therefore, the correct instrument for drawing a circle is the Compass.

The correct option is (C).

When using a compass to draw a circle with a specific radius, one arm of the compass, which usually has a sharp point, is fixed at a particular location. This fixed point represents the centre of the circle.

The other arm, holding a pencil or pen, is then rotated around this fixed point while maintaining a constant distance from it. This constant distance is the radius of the circle.

Let's consider the other options:

(A) Circumference: The circumference is the boundary line of the circle itself, not the point around which it is drawn.

(B) Radius: The radius is a distance, not a point.

(D) Diameter: The diameter is a line segment passing through the centre, or its length, not the central point.

Therefore, the sharp point of the compass is placed at the Centre to draw a circle of a given radius.

The correct option is (B).

A Ruler is the standard geometrical instrument used for drawing straight lines and measuring lengths.

To draw a line segment of a specific length, such as $5.5$ cm, you simply use the ruler to mark the starting point, then draw a straight line along the edge of the ruler up to the $5.5$ cm mark.

Let's look at the other options:

(A) Compass and protractor: A compass is for drawing circles, and a protractor is for angles. Neither is primarily used for drawing a straight line segment of a specific length.

(C) Protractor and ruler: While a ruler is needed, the protractor is for angles and is not required to draw a simple line segment of a given length.

(D) Compass and ruler: A ruler is needed, but a compass is for drawing circles or arcs, not straight line segments.

Therefore, the most appropriate and sufficient instrument to draw a line segment of a given length is a Ruler only.

The correct option is (C).

A Divider is the most suitable instrument for comparing the lengths of two line segments accurately without the need to read measurements from a scale (using numbers).

A divider has two sharp points. To compare two line segments, you can open the divider to match the length of the first segment by placing its points on the endpoints of the segment. Then, keeping the opening of the divider the same, place its points on the endpoints of the second segment. By observing if the second segment is shorter than, equal to, or longer than the opening of the divider, you can compare their lengths without explicitly measuring them with numbers.

Let's consider why other options are less suitable for this specific task:

(A) Ruler: A ruler is used to measure length, which involves reading numbers from a scale. The question asks for comparison *without using numbers*.

(B) Protractor: A protractor is used for measuring and drawing angles, which is irrelevant to comparing line segment lengths.

(D) Compass: While a compass can be used to transfer distances (similar to a divider), a divider is specifically designed for comparison and transfer of lengths and often provides better stability and precision for this task, especially when comparing segments directly without drawing arcs.

Therefore, the Divider is the best instrument for comparing the lengths of two line segments accurately without using numerical measurements.

The correct option is (C).

The construction of a perpendicular to a line through a point on the line using a compass typically begins by positioning the compass.

The steps are:

1. Place the compass at the point on the line through which the perpendicular is to be drawn. This point will be the center for the initial arcs.

2. With the compass point at the given point, open the compass to a convenient radius and draw arcs intersecting the line on both sides of the point.

3. Open the compass to a radius greater than the distance from the point to either intersection point. With the compass point at each intersection point, draw arcs above (or below) the line such that they intersect each other.

4. Draw a straight line segment connecting the original point on the line to the intersection point of the two arcs drawn in step 3. This line segment is perpendicular to the original line.

Looking at the options provided:

(A) Draw arcs intersecting the line on both sides of the point: This is the second step.

(B) Draw a line segment: While you start with a line segment (or line) and a point on it, drawing *a* line segment is not the first step of the construction process *using a compass* once the line and point are given.

(C) Place the compass at the point: This is the initial action taken with the compass, placing its pivot point at the specified location on the line.

(D) Draw a line perpendicular to the given line: This is the final outcome of the construction, not the first step.

Therefore, the first action when using a compass for this construction is to Place the compass at the point.

The correct option is (B).

A perpendicular bisector of a line segment is a line that satisfies two conditions:

1. It is perpendicular to the line segment, meaning it intersects the segment at a $90^\circ$ angle.

2. It bisects the line segment, meaning it passes through the midpoint of the segment, thereby dividing it into two equal parts.

Let's examine the given options:

(A) Divides the segment into two unequal parts: This is incorrect. A bisector always divides the segment into two equal parts.

(B) Forms a $90^\circ$ angle with the segment: This is correct. This is the definition of the "perpendicular" part of a perpendicular bisector.

(C) Passes through one endpoint of the segment: This is incorrect. A bisector passes through the midpoint of the segment, which is generally different from the endpoints (unless the segment is of zero length).

(D) Only forms a $90^\circ$ angle, but doesn't divide equally: This is incorrect. A perpendicular bisector by definition does both: it forms a $90^\circ$ angle *and* divides the segment equally.

Among the given options, only (B) is a correct property of a perpendicular bisector.

The correct option is (B).

Using only a compass and a ruler, we can construct certain angles. Fundamental constructible angles include $60^\circ$ and $90^\circ$. Other angles can be constructed by bisecting constructible angles or by adding or subtracting constructible angles.

Let's examine the options:

(A) $10^\circ$: This angle is not constructible using only a compass and ruler. Constructing $10^\circ$ would require trisecting a $30^\circ$ angle ($30^\circ/3 = 10^\circ$), and trisecting a general angle is not possible with only compass and ruler.

(B) $45^\circ$: This angle is constructible. We can construct a $90^\circ$ angle (a perpendicular line) and then bisect it using the compass and ruler. Bisecting a $90^\circ$ angle yields two $45^\circ$ angles.

(C) $70^\circ$: This angle is not constructible using only a compass and ruler. Angles that are integer multiples of $10^\circ$ (like $10^\circ, 20^\circ, 40^\circ, 50^\circ, 70^\circ, 80^\circ, 100^\circ$, etc., excluding multiples of $15^\circ$) are generally not constructible.

(D) $80^\circ$: This angle is also not constructible using only a compass and ruler.

Angles that can be constructed using compass and ruler are those whose measure can be expressed in the form $\frac{360^\circ}{2^n \cdot F_{m_1} \cdots F_{m_k}}$, where $F_{m_i}$ are distinct Fermat primes ($3, 5, 17, 257, 65537$). This includes angles that are integer multiples of $3^\circ$ only if they are also integer multiples of $15^\circ$ or $22.5^\circ$ etc., obtainable through these methods.

Since $45^\circ = \frac{180^\circ}{4} = \frac{360^\circ}{8}$, which is of the form $\frac{360^\circ}{2^3}$, it is constructible.

Therefore, out of the given options, only $45^\circ$ can be constructed using a compass and ruler.

The correct option is (C).

To construct an angle of $90^\circ$, we need to divide an angle into two equal parts where each part is $90^\circ$. Bisecting an angle means dividing it into two equal halves.

We know that:

$90^\circ + 90^\circ = 180^\circ$

An angle measuring $180^\circ$ is called a straight angle because it forms a straight line.

Bisecting a straight angle ($180^\circ$) results in two angles, each measuring half of $180^\circ$, which is $90^\circ$. This is the standard method for constructing a $90^\circ$ angle using a compass and ruler, often done by constructing the perpendicular bisector of a line segment or by constructing a perpendicular to a line at a point on the line, which involves creating two $90^\circ$ angles on either side of the perpendicular line.

Let's consider the other options:

(A) Acute angle: Bisecting an acute angle (less than $90^\circ$) results in an angle less than $45^\circ$. This does not give $90^\circ$.

(B) Obtuse angle: Bisecting an obtuse angle (greater than $90^\circ$ but less than $180^\circ$) results in an angle greater than $45^\circ$ but less than $90^\circ$. This does not give $90^\circ$.

(D) Reflex angle: Bisecting a reflex angle (greater than $180^\circ$ but less than $360^\circ$) results in an angle greater than $90^\circ$. This does not give $90^\circ$.

Therefore, bisecting a straight angle ($180^\circ$) is the way to obtain a $90^\circ$ angle.

The correct option is (D).

The geometrical instrument specifically designed for measuring the size of angles and drawing angles of specific measures is a Protractor.

A protractor is typically a semicircular or circular tool marked with degrees along its edge. To measure an angle, the center of the protractor is placed on the vertex of the angle, and the baseline of the protractor is aligned with one arm of the angle. The measure of the angle is then read from the scale where the other arm of the angle intersects the protractor's edge.

To draw an angle of a specific measure, a point is marked for the vertex, and a line segment for one arm. The protractor is placed with its center on the vertex and baseline along the first arm. A mark is made at the desired degree measure on the protractor's scale, and a line segment is drawn from the vertex through this mark.

Let's consider the other options:

(A) Compass: Used for drawing circles and arcs.

(B) Ruler: Used for drawing straight lines and measuring lengths.

(C) Divider: Used for transferring or comparing lengths.

None of these are used for measuring or drawing angles in the way a protractor is.

Therefore, the Protractor is the instrument used to measure and draw angles accurately.

The correct option is (B).

To construct an angle equal to a given angle without using a protractor, we rely on geometric constructions using only a Compass and a Ruler.

The standard method involves the following steps:

1. Draw a ray which will be one arm of the new angle using a ruler.

2. Place the compass point at the vertex of the given angle and draw an arc that intersects both arms of the given angle.

3. Without changing the compass setting, place the compass point at the endpoint of the new ray and draw a similar arc that intersects the ray.

4. Measure the distance between the two points where the arc intersects the arms of the given angle using the compass.

5. With this new compass setting, place the compass point at the point where the second arc intersects the new ray, and draw an arc that intersects the second arc.

6. Draw a ray from the endpoint of the first ray through the intersection point of the two arcs using a ruler. This ray forms the second arm of the new angle, which is equal to the given angle.

As you can see, this construction exclusively uses a compass (for drawing arcs and transferring distances) and a ruler (for drawing rays/lines).

Let's consider the other options:

(A) Ruler and protractor: The question specifies "without using a protractor".

(C) Divider and ruler: A divider is primarily for comparing or transferring lengths, not for accurately transferring angle measures in this construction method.

(D) Only a protractor: The question specifies "without using a protractor".

Therefore, the instruments used to construct an angle equal to a given angle without a protractor are a Compass and ruler.

The correct option is (B).

A perpendicular bisector is a line that intersects a line segment at its midpoint and is perpendicular to it.

The term "perpendicular" means that the line forms a right angle with the segment it intersects. A right angle measures $90^\circ$.

When a line intersects another line (or segment) at a point, it creates angles around that point. If the intersection is perpendicular, the angle formed is $90^\circ$. A line crossing a segment forms two straight angles ($180^\circ$) on either side of the intersection point. The perpendicular bisector divides each of these $180^\circ$ straight angles into two $90^\circ$ angles.

Therefore, the angles formed by the perpendicular bisector with the line segment at the point of intersection are two $90^\circ$ angles on each side of the bisector along the segment.

Let's look at the options:

(A) Two $45^\circ$ angles: Incorrect. $45^\circ$ angles are formed by bisecting a $90^\circ$ angle, not by a perpendicular bisector itself (unless it's bisecting a $90^\circ$ angle, which isn't the primary definition here).

(B) Two $90^\circ$ angles: Correct. This is the definition of being perpendicular.

(C) One $90^\circ$ and one $180^\circ$ angle: Incorrect. While a $180^\circ$ angle is involved in the understanding of bisecting a straight angle to get $90^\circ$, the angles directly formed by the intersection of the perpendicular bisector and the segment are $90^\circ$.

(D) Two equal acute angles: Incorrect. Acute angles are less than $90^\circ$. The angles are equal, but they are $90^\circ$, which are right angles, not acute angles.

Thus, a perpendicular bisector forms two $90^\circ$ angles with the line segment at the point of bisection.

The correct option is (A).

An angle of $60^\circ$ is one of the fundamental angles that can be easily constructed using only a compass and ruler (by drawing an equilateral triangle or by using arcs of the same radius centered at two points on a line). Once a $60^\circ$ angle is constructed, to obtain a $30^\circ$ angle, you need to divide the $60^\circ$ angle into two equal parts.

The geometric operation of dividing an angle into two equal angles is called bisecting the angle.

Mathematically, $60^\circ \div 2 = 30^\circ$. So, dividing the $60^\circ$ angle by two gives the required $30^\circ$ angle.

Let's consider the options:

(A) Bisect: This means dividing the angle into two equal parts. Bisecting $60^\circ$ gives $30^\circ$. This is correct.

(B) Triple: Tripling $60^\circ$ would give $60^\circ \times 3 = 180^\circ$. This is incorrect.

(C) Halve: This also means dividing the angle into two equal parts, which is synonymous with bisecting in this context. While mathematically correct, "Bisect" is the more standard geometric term for the construction method.

(D) Extend: Extending the arms of an angle does not change the measure of the angle. This is incorrect.

Therefore, after constructing a $60^\circ$ angle, you would Bisect it to get a $30^\circ$ angle using a compass and ruler.

The correct option is (B).

When constructing a perpendicular to a given line from a point outside the line using a compass, the first step is crucial for establishing reference points on the line.

Given a line $l$ and a point $P$ not on $l$.

The construction steps begin by placing the compass point at the external point $P$. Then, with the compass open to a suitable radius (large enough to cross the line), an arc is drawn. This arc should intersect the given line $l$ at two distinct points.

Let's analyze the options based on this first step:

(A) They should be of different radii: The initial arc(s) drawn from the external point typically use a single radius to intersect the line at two points. While different radii might be used for subsequent arcs, the initial arcs are usually drawn with one setting.

(B) They should be drawn from the point outside the line: This is correct. The compass point is placed at the given external point, and the arc(s) are drawn from there to intersect the line.

(C) They should intersect each other above the line: This refers to the *second* set of arcs drawn from the points on the line, which intersect each other to help locate the path of the perpendicular. The *first* arcs intersect the line itself.

(D) They should be drawn from points on the line: This describes the origin of the *second* set of arcs used in the construction, not the initial arcs that first interact with the line from the external point.

Therefore, the defining characteristic of the first arcs drawn in this construction is that they originate from the point outside the line.

The angles that can be constructed using a compass and ruler are those which can be expressed in the form $\frac{360^\circ}{2^n \cdot F_{m_1} \cdots F_{m_k}}$, where $F_{m_i}$ are distinct Fermat primes ($3, 5, 17, 257, 65537$). Angles obtained by bisecting constructible angles or by summing/subtracting constructible angles are also constructible.

Let's analyze each option:

(A) $15^\circ$: We can construct a $60^\circ$ angle. Bisecting $60^\circ$ gives $30^\circ$. Bisecting $30^\circ$ gives $15^\circ$. Alternatively, construct $60^\circ$ and $90^\circ$. The angle between them is $30^\circ$. Bisecting $30^\circ$ gives $15^\circ$. So, $15^\circ$ is constructible by repeatedly bisecting $60^\circ$ or derived angles.

(B) $22.5^\circ$: We can construct a $90^\circ$ angle. Bisecting $90^\circ$ gives $45^\circ$. Bisecting $45^\circ$ gives $22.5^\circ$. So, $22.5^\circ$ is constructible by repeatedly bisecting $90^\circ$.

(C) $75^\circ$: We can construct $60^\circ$. We can construct $15^\circ$ (as shown in A). $75^\circ = 60^\circ + 15^\circ$. Since $60^\circ$ is constructible and $15^\circ$ is constructible (derived from bisections), their sum is also constructible.

(D) $105^\circ$: We can construct $60^\circ$ and $45^\circ$ (as shown in B). $105^\circ = 60^\circ + 45^\circ$. Since both $60^\circ$ and $45^\circ$ are constructible (where $45^\circ$ is derived from bisection), their sum is also constructible.

All listed angles ($15^\circ, 22.5^\circ, 75^\circ, 105^\circ$) are constructible using a compass and ruler, and their construction methods involve either direct repeated bisection of $60^\circ$ or $90^\circ$, or combining angles obtained through such bisections with the standard $60^\circ$ and $90^\circ$ angles.

Therefore, all options are correct.

The correct options are (A), (B), (C), and (D).

The correct option is (A).

Let's analyze the Assertion and the Reason:

Assertion (A): A circle has only one centre.

This statement is true. By definition, a circle is defined relative to a single fixed point from which all points on the circle are equidistant.

Reason (R): A circle is the set of all points in a plane that are equidistant from a fixed point, called the centre.

This statement is also true. This is the standard definition of a circle.

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

The reason (R) defines a circle based on its relationship to a fixed point, which is explicitly called the centre. This definition inherently implies that there is only one such fixed point from which all points on the circle are equidistant at a constant distance (the radius).

If a circle had more than one "centre", points on the circle would need to be simultaneously equidistant from multiple distinct points, which is impossible unless the "circle" is just a single point or the multiple "centres" are actually the same point.

Therefore, the definition provided in Reason (R) directly explains why a circle must have only one centre (Assertion A).

Both the Assertion and the Reason are true, and the Reason correctly explains the Assertion.

The correct option is (C).

The carpenter needs to mark the diagonal of the rectangular plank. A diagonal is a straight line connecting two opposite corners (points) of the rectangle.

Let's consider the options:

(A) Drawing a circle: This involves creating a curved shape, not a straight line between two specific points.

(B) Constructing a perpendicular: This is a specific construction to create a line at a $90^\circ$ angle to another line. While it results in a straight line, it's not the general concept of drawing any straight line between two points.

(C) Drawing a line segment: A line segment is defined as a straight line connecting two distinct points. This is precisely what the carpenter needs to do - draw a straight line between the two corner points to mark the diagonal.

(D) Measuring an angle: This involves finding the size of an angle and does not directly relate to the action of drawing a straight line.

Therefore, the geometrical action most directly related to drawing a straight line between two points is Drawing a line segment. The tool used for this is typically a ruler or a straightedge.

The correct option is (B).

The center of a rectangle is the point where its diagonals intersect. It is also the intersection point of the lines that are perpendicular to the sides and pass through their midpoints. These lines are known as the perpendicular bisectors of the sides.

To find the center of the rectangular plank using a method related to perpendicular lines, the carpenter can:

1. Find the midpoint of one side of the rectangle.

2. Construct a line perpendicular to that side, passing through its midpoint (i.e., construct the perpendicular bisector of that side).

3. Repeat steps 1 and 2 for an adjacent side.

4. The point where these two perpendicular bisectors intersect is the exact center of the rectangle.

This method involves constructing perpendicular lines that also bisect the sides.

Let's look at the other options:

(A) Drawing an angle bisector: Angle bisectors divide angles and are used, for example, to find the incenter of a triangle. They are not used to find the center of a rectangle.

(C) Drawing parallel lines: While rectangles have parallel sides, drawing parallel lines in general doesn't directly help in finding the center unless they are specific lines related to the midpoints or diagonals.

(D) Drawing a tangent: Tangents are lines that touch curves at a single point, typically associated with circles. They are not relevant to finding the center of a rectangle.

Therefore, the concept of drawing a perpendicular bisector of the sides is a useful geometrical concept related to perpendicular lines for finding the exact center of a rectangular plank.

The correct option is (A).

Let's match each construction/action with the appropriate tool(s) or method:

(i) Drawing an angle of $70^\circ$: Angles of arbitrary measures like $70^\circ$ are typically drawn and measured using a protractor along with a ruler to draw the rays. This corresponds to (b) Protractor and Ruler.

(ii) Copying a line segment length: This involves transferring a specific distance from one location to another. While a compass and ruler can be used (by opening the compass to the length and marking an arc), a divider is specifically designed for this purpose (transferring lengths). Using a compass and a divider is a direct way to copy a length. This corresponds to (c) Compass and Divider.

(iii) Constructing $60^\circ$ angle: The construction of a $60^\circ$ angle is a fundamental geometric construction that is performed using only a compass and a ruler. This corresponds to (a) Compass and Ruler.

(iv) Dividing a line segment equally into two parts: This is the process of finding the midpoint of a line segment. The standard method for this construction is the perpendicular bisector construction, which guarantees the point found is the midpoint and the line is perpendicular. This corresponds to (d) Perpendicular Bisector Construction.

Matching the pairs:

(i) $\to$ (b)

(ii) $\to$ (c)

(iii) $\to$ (a)

(iv) $\to$ (d)

This sequence matches option (A).

The correct option is (A).

The method for drawing a perpendicular to a line at a given point P on the line using a set square and a ruler involves using the right-angle property of the set square.

Here is the standard procedure:

1. Place the ruler along the given line.

2. Place the set square such that one of the edges forming the right angle ($90^\circ$) rests firmly against the edge of the ruler, and the vertex of the right angle is positioned exactly at the point P on the line.

3. Hold both the ruler and the set square steady.

4. Draw a line segment or a ray along the other edge of the set square that forms the right angle, starting from point P. This line will be perpendicular to the original line at point P.

Let's evaluate the options based on this procedure:

(A) Align the ruler with the line, place the set square's right angle vertex at P, and draw the perpendicular along the edge of the set square: This correctly describes the steps involved in using a set square and ruler to draw a perpendicular at a point on the line.

(B) Align the set square with the line and draw a line along the ruler: This action would typically result in drawing a line parallel to the set square's aligned edge, possibly using the ruler as a guide, but it doesn't guarantee a perpendicular at point P.

(C) Draw an arc with P as centre: This is the initial step in constructing a perpendicular using a compass and ruler, not a set square and ruler.

(D) Measure $90^\circ$ with a protractor: This describes the method using a protractor, not a set square and ruler.

Therefore, the correct method using a set square and ruler is described in option (A).

The correct option is (B).

To construct any angle using a compass and ruler, you need a starting point and a baseline. This baseline is typically represented by a ray.

The standard steps for constructing a $60^\circ$ angle are:

1. Draw a ray, say OA. This ray will form one arm of the angle, and the endpoint O will be the vertex.

2. With O as the center and a convenient radius, draw an arc intersecting the ray OA at a point, say P.

3. With P as the center and the same radius (the one used in step 2), draw another arc that intersects the first arc at a point, say Q.

4. Draw a ray OQ. The angle $\angle$AOQ is a $60^\circ$ angle.

Let's consider the given options:

(A) Draw an arc from the vertex intersecting the ray: This is the second step, done after the ray is drawn and the vertex is established.

(B) Draw a ray: This is the necessary initial step to provide the base for the angle construction.

(C) Draw a circle: Drawing a full circle is not the required first step; only an arc is needed initially.

(D) Mark a point on the ray: A point on the ray is marked in step 2, after the ray is drawn.

Therefore, the very first step is to Draw a ray.

The correct option is (D).

The construction of a perpendicular bisector of a line segment using a compass and ruler involves the following steps:

1. Draw a line segment. (This is the segment you want to bisect perpendicularly).

2. With the compass point placed at one endpoint of the segment, open the compass to a radius that is greater than half the length of the segment. Draw arcs above and below the segment.

3. With the compass point placed at the other endpoint of the segment, and using the same radius as in step 2, draw arcs above and below the segment. These arcs should intersect the previously drawn arcs at two distinct points.

4. Using a ruler, join the points where the arcs intersect. This line segment is the perpendicular bisector of the original line segment.

Reviewing the options:

(A) Draw a line segment: This is the object on which the construction is performed, a necessary starting point.

(B) With radius greater than half the length of the segment, draw arcs from both endpoints: This is a crucial step in the construction process to locate the points through which the bisector will pass.

(C) Join the points where the arcs intersect: This is the final step to draw the perpendicular bisector line.

(D) Measure the angle formed at the midpoint using a protractor: This step is typically done *after* the construction to verify that the angle formed is $90^\circ$, confirming that the line is indeed perpendicular. However, it is not a step *in* the construction process itself when using the compass and ruler method. The method geometrically ensures perpendicularity.

Therefore, measuring the angle with a protractor is not a required step for constructing a perpendicular bisector using compass and ruler.

The correct option is (B).

When drawing a circle using a compass, one arm of the compass has a sharp point that is placed at a fixed position on the paper or surface. This fixed point acts as the pivot around which the other arm (with the pencil or pen) rotates to trace the circular path.

By definition, this fixed point, from which all points on the circle are equidistant, is called the Centre of the circle.

Let's look at the other options:

(A) Vertex: A vertex is typically the point where two or more lines or edges meet, often associated with angles or polygons.

(C) Radius: The radius is the distance from the centre to any point on the circle, not the fixed point itself.

(D) Arc: An arc is a portion of the circumference of a circle, which is drawn by the moving arm of the compass, not the fixed point.

Therefore, the point where the compass is fixed while drawing a circle is called the Centre.

The correct option is (B).

Bisecting an angle means dividing it into two equal angles.

If we bisect a $90^\circ$ angle, we divide the angle measure by 2.

Measure of each resulting angle $= \frac{90^\circ}{2} = 45^\circ$.

So, the resulting angles will each measure $45^\circ$.

Measure = $45^\circ$

The correct option is (D).

Let's examine the shapes and features of the given geometrical instruments:

(A) Ruler: A ruler is typically a straight strip used for measuring length and drawing straight lines. It does not have a defined right angle at a corner for construction purposes (though the corner itself is $90^\circ$, it's the edge that is used for drawing lines).

(B) Protractor: A protractor is usually semicircular or circular and is used for measuring and drawing angles. While it has various angle markings, it doesn't have a designated right angle *at a corner* used for drawing perpendiculars.

(C) Compass: A compass is used for drawing circles and arcs. It consists of two arms joined at a pivot and does not have corners with specific angles.

(D) Set square: Set squares are triangular tools specifically designed with precise angles for drawing lines, especially perpendicular and parallel lines. Common set squares have angles of $90^\circ, 45^\circ, 45^\circ$ or $90^\circ, 30^\circ, 60^\circ$. Thus, a set square has a right angle ($90^\circ$) at one of its corners.

Therefore, the instrument among the options that specifically has a right angle at one of its corners, used as a tool for construction, is a Set square.

The correct option is (A).

To copy a line segment AB to a new location using only a compass and ruler, the standard construction method is as follows:

1. First, draw a ray (a line starting from a point and extending infinitely in one direction) using a ruler. This ray will serve as the base for the copied segment.

2. Next, open the compass to the length of the original line segment AB. This is done by placing the sharp point of the compass at point A and adjusting the pencil tip to rest exactly on point B.

3. Now, place the sharp point of the compass at the starting point of the new ray you just drew.

4. Keeping the compass opening (radius) the same as the length of AB, draw an arc that intersects the ray.

5. The point where the arc intersects the ray is the endpoint of the copied segment. The segment from the start of the ray to this intersection point will have the exact same length as the original segment AB.

Let's look at why the other options are not the standard compass and ruler copying method:

(B) Measure AB with a ruler and then draw a new segment of that length: While this uses both tools, the question implies a geometric construction method of copying without necessarily relying on reading the numerical measurement from the ruler.

(C) Trace the line segment AB: Tracing is a physical copying method, not a geometric construction using compass and ruler.

(D) Draw an arc with A as centre and B as radius: This action is part of drawing a circle around point A, and it doesn't directly result in a copy of the segment AB itself located elsewhere.

Therefore, option (A) correctly describes the process of copying a line segment using a compass and ruler.

The correct option is (C).

Let's define the terms given in the options:

(A) Angle bisector: A line or ray that divides an angle into two equal angles. It does not relate to dividing a line segment or being perpendicular to it.

(B) Perpendicular from a point on the line: A line drawn at $90^\circ$ to a given line, passing through a specific point *on* that line. It creates perpendicularity but does not necessarily bisect any segment.

(C) Perpendicular bisector: A line that is perpendicular to a given line segment and passes through its midpoint, thereby dividing it into two equal parts. This definition perfectly matches the description in the question.

(D) Copying a line segment: The process of creating a new line segment of the same length as a given segment at a different location. This construction does not involve perpendicularity or bisection of the original segment.

The construction that results in a line that is both perpendicular to a given line segment and divides it into two equal parts is the Perpendicular bisector.

The correct option is (B).

When an angle is bisected, it is divided into two equal parts. Each part has half the measure of the original angle.

We start with a $60^\circ$ angle.

First bisection:

Bisecting the $60^\circ$ angle results in two angles.

Measure of each angle after the first bisection $= \frac{60^\circ}{2} = 30^\circ$.

So, we now have angles of $30^\circ$.

Second bisection:

We then bisect the resulting angles. If we bisect one of the $30^\circ$ angles, it is divided into two equal parts.

Measure of each angle after the second bisection $= \frac{30^\circ}{2} = 15^\circ$.

After bisecting the original $60^\circ$ angle twice, we end up with angles that each measure $15^\circ$.

The smallest angle obtained is $15^\circ$.

The correct option is (B).

A compass is primarily used for drawing circles and arcs. The distance between the sharp point (pivot) and the pencil tip of a compass determines the radius of the circle or arc being drawn.

Let's look at the options:

(A) Drawing any line: Lines are drawn using a ruler or a straightedge. A compass is not used for drawing straight lines.

(B) Drawing a circle of a specific radius: This is the primary function of a compass. To draw a circle of a specific radius, you must set the distance between the compass points equal to that desired radius. This requires setting a specific distance on the compass.

(C) Drawing any angle: Angles are typically measured and drawn using a protractor. While a compass is used in the construction of specific angles (like $60^\circ, 90^\circ$, etc.) by drawing arcs of certain radii, drawing *any* arbitrary angle to a specific measure requires a protractor. Moreover, the compass radius in angle constructions is often chosen for convenience, not necessarily set to a specific predetermined numerical distance unless copying an angle.

(D) Measuring a line segment with a ruler: This is done using a ruler's marked scale. While a compass can be used to transfer a length, measuring involves reading a number from the ruler, which does not require setting a specific distance *on the compass* first (you set the compass based on the length you measure).

Therefore, drawing a circle of a specific radius directly requires setting a specific distance (that radius) on the compass.

The correct option is (C).

The case study describes the architect needing to draw a new wall that is at a right angle ($90^\circ$) to an existing wall and starts from a specific point.

Let's analyze the options in the context of the description:

(A) Copying a line segment: This construction is about duplicating a length, which is not the primary task described.

(B) Drawing a circle: This construction is about creating a circular shape, which is not needed for drawing walls in this context.

(C) Constructing a perpendicular: This is the process of drawing a line that forms a $90^\circ$ angle with another line, often through a given point. Drawing a wall "perpendicular to an existing wall from a specific point" is precisely the definition of constructing a perpendicular line from a point on the existing wall's line or from a point outside it, depending on where the "specific point" is located relative to the existing wall's line.

(D) Bisecting an angle: This construction divides an angle into two equal parts, which is not relevant to drawing a wall perpendicular to another.

Therefore, the geometrical construction the architect is performing is Constructing a perpendicular.

The correct option is (B).

The architect needs to divide a planned wall segment into two equal halves. This means finding the exact midpoint of the segment.

Let's look at the provided options:

(A) Constructing a $90^\circ$ angle: This construction creates a right angle but does not necessarily divide a line segment into equal parts.

(B) Constructing a perpendicular bisector: A perpendicular bisector is a line that is perpendicular to a segment and passes through its midpoint. The term "bisector" specifically means dividing the segment into two equal parts. The process of constructing a perpendicular bisector is the standard geometric method using compass and ruler to find the midpoint of a line segment and draw a line through it that is perpendicular to the segment.

(C) Drawing parallel lines: This involves creating lines that never intersect and maintain a constant distance from each other. This construction is not used for dividing a line segment.

(D) Measuring with a protractor: A protractor is used for measuring and drawing angles, not for finding the midpoint of a line segment.

Therefore, the most useful construction for dividing a planned wall segment into two equal halves is Constructing a perpendicular bisector, as it specifically locates the midpoint of the segment.

The correct option is (C).

Let's evaluate each statement regarding constructions using only a compass and ruler:

(A) A $60^\circ$ angle is a fundamental construction: This statement is TRUE. Constructing a $60^\circ$ angle is one of the basic constructions using a compass and ruler.

(B) You can construct angles like $30^\circ$ and $120^\circ$: This statement is TRUE. A $30^\circ$ angle can be constructed by bisecting a $60^\circ$ angle. A $120^\circ$ angle can be constructed by doubling a $60^\circ$ angle or by extending one arm of a $60^\circ$ angle and noting that $120^\circ$ is the supplementary angle to $60^\circ$ on a straight line ($180^\circ - 60^\circ = 120^\circ$), or by constructing two adjacent $60^\circ$ angles.

(C) You can construct any angle measure exactly: This statement is FALSE. As discussed in a previous question, only certain angles are constructible using a compass and ruler. For example, $10^\circ, 20^\circ, 70^\circ, 80^\circ$ are generally not constructible. The ability to construct an angle is related to algebraic properties of the cosine of the angle.

(D) Angle bisectors are used to construct angles like $45^\circ$ and $22.5^\circ$: This statement is TRUE. A $45^\circ$ angle is obtained by bisecting a $90^\circ$ angle. A $22.5^\circ$ angle is obtained by bisecting a $45^\circ$ angle.

The statement that is FALSE is (C).

The correct option is (C).

The construction of a perpendicular bisector of a line segment involves drawing arcs from the two endpoints of the segment.

Let the line segment be AB.

We place the compass point at A and draw arcs above and below the segment. Then, keeping the same compass radius, we place the compass point at B and draw arcs above and below the segment.

For the arcs from A and B to intersect at two distinct points (one on each side of the segment), the radius used for drawing these arcs must be large enough to ensure overlap.

If the radius were exactly half the length of AB, the arcs from A and B would meet at exactly one point, the midpoint of AB. While this point lies on the perpendicular bisector, the method relies on two intersection points to define the line.

If the radius were less than half the length of AB, the arcs drawn from A and B would not intersect at all.

Therefore, the radius used must be greater than half of AB to guarantee that the arcs intersect at two points.

These two intersection points define the line that is the perpendicular bisector of AB.

The correct option is (B).

Let the given line be $l$ and the point outside the line be P.

The first step is as described: place the compass point at P and draw an arc that intersects the line $l$ at two points, say A and B.

These two points A and B on the line are now equidistant from P. The next step is to find another point (or points) that are equidistant from A and B.

The standard construction method continues as follows:

1. With center A and a radius greater than half the length of segment AB, draw an arc on the side of the line $l$ opposite to point P.

2. With center B and the same radius as used in step 1, draw another arc that intersects the arc drawn from A. Let the intersection point be Q.

3. The line segment PQ is the perpendicular to line $l$ from point P.

Looking at the options after having found the intersection points A and B on the line:

(A) Draw a line through the point and one intersection: Drawing PA or PB does not result in a perpendicular line to $l$ unless P happens to lie on the perpendicular bisector of some segment on the line.

(B) With centres at the two intersection points, draw arcs of the same radius on the opposite side of the line: This accurately describes the next crucial step in the standard construction method. The radius used from A and B should be the same and large enough for the arcs to intersect.

(C) Measure the angle at the intersection points: Measuring angles is done with a protractor and is not a step in the compass and ruler construction process itself.

(D) Draw a circle with the outside point as centre: The first step involves drawing *an arc* from the outside point, not necessarily a full circle. Drawing a full circle isn't the specific next step to construct the perpendicular.

Therefore, the next major step is to draw intersecting arcs from the two points found on the line.

The correct option is (C).

In geometry, certain angles can be constructed accurately using only a compass and ruler. These include angles like $60^\circ$ (a fundamental construction), $90^\circ$ (by constructing a perpendicular), and angles derived from these by bisection or combination (like $30^\circ, 45^\circ, 120^\circ$).

Let's examine the options:

(A) $90^\circ$: This angle can be constructed accurately using a compass and ruler (by drawing a perpendicular).

(B) $60^\circ$: This angle can be constructed accurately using a compass and ruler (a fundamental construction).

(C) $55^\circ$: This angle measure is not one of the standard angles that can be easily constructed using only a compass and ruler through simple bisections or combinations of $60^\circ$ or $90^\circ$. To draw an angle like $55^\circ$ accurately, especially in Class 6 where complex constructions might not be taught, a protractor is the standard tool.

(D) $120^\circ$: This angle can be constructed accurately using a compass and ruler ($180^\circ - 60^\circ$ or $60^\circ + 60^\circ$).

Therefore, among the given options, $55^\circ$ is the angle that would most likely require a protractor to be drawn accurately in Class 6.

The correct option is (B).

Practical geometry focuses on performing geometric constructions accurately using specific tools.

The fundamental tools used in classical geometric constructions are the ruler (or straightedge, for drawing straight lines) and the compass (for drawing circles and arcs, and transferring distances).

The sentence mentions the ruler and asks for the other key tool.

Let's examine the options:

(A) Protractor: Used for measuring and drawing angles, but compass and ruler constructions are a distinct set of methods.

(B) Compass: Used for drawing circles and arcs, essential for transferring lengths and creating intersections in most constructions.

(C) Divider: Used for comparing and transferring lengths, often a specialized version of a compass, but compass is the more general construction tool.

(D) Set square: Used for drawing specific angles (like $90^\circ$) and parallel lines, often used with a ruler, but compass and ruler are the core pair for a wider range of constructions.

Therefore, practical geometry extensively uses the ruler and the Compass for geometric constructions.

Tool: Compass

The geometric instrument commonly used to draw circles is a compass or a pair of compasses.

This instrument typically consists of two legs joined at a hinge. One leg has a sharp point to fix the center of the circle, and the other leg holds a pencil or pen to draw the circumference as the instrument rotates around the fixed point.

To draw a circle, you need two key pieces of information:

1. The location of the center of the circle. This is the fixed point around which all points on the circle's circumference are equidistant.

2. The length of the radius of the circle. This is the fixed distance from the center to any point on the circle's circumference.

Knowing these two pieces of information allows you to uniquely define and draw a circle.

The first step in constructing a line segment of a given length using a ruler is to mark a point on your paper.

This marked point will serve as the starting point (one endpoint) of your line segment.

You typically align the zero mark of the ruler with this starting point.

To compare the lengths of two line segments using a compass, follow these steps:

1. Measure the first line segment:

Place the sharp point of the compass on one endpoint of the first line segment.

Adjust the opening of the compass so that the pencil end rests exactly on the other endpoint of the same line segment.

2. Transfer the measurement to the second line segment:

Without changing the fixed opening of the compass, lift it and place the sharp point on one endpoint of the second line segment.

3. Compare the lengths:

Observe the position of the pencil end of the compass relative to the second endpoint of the second line segment.

If the pencil end falls exactly on the second endpoint, the two line segments are of equal length.

If the pencil end falls short of the second endpoint, the first line segment (the one measured with the compass) is shorter than the second line segment.

If the pencil end falls beyond the second endpoint, the first line segment is longer than the second line segment.

This method provides a direct comparison of lengths without needing to read specific measurements from a ruler.

Perpendicular lines are two lines that intersect each other at a right angle.

A right angle measures $90^\circ$.

The symbol for perpendicularity between two lines, say line $L_1$ and line $L_2$, is $L_1 \perp L_2$.

A perpendicular bisector of a line segment is a line that satisfies two conditions:

1. It is perpendicular to the line segment.

This means it intersects the line segment at a right angle ($90^\circ$).

2. It bisects the line segment.

This means it passes through the midpoint of the line segment, dividing it into two equal parts.

Therefore, a perpendicular bisector is a line that intersects a given line segment at its midpoint and forms a $90^\circ$ angle with it.

The geometric tool essential for constructing a perpendicular using arcs is a compass.

A compass is used to draw arcs of specific radii from given points, which is a fundamental step in the standard construction methods for perpendicular lines (e.g., constructing a perpendicular bisector or constructing a perpendicular from a point to a line or through a point on a line).

Perpendicular lines intersect at a right angle.

A right angle measures $90^\circ$.

The first step to construct a perpendicular to a line through a point on the line using a compass is to place the compass point on the given point on the line.

Then, with any convenient radius (usually greater than zero but not too large), draw arcs that intersect the line on both sides of the given point.

The first step to construct a perpendicular from a point outside a line to the line using a compass is to place the compass point on the given external point.

Then, with a radius large enough to intersect the line, draw an arc that crosses the given line at two distinct points.

The instrument used to measure angles is called a protractor.

A protractor is usually a semicircular or circular tool with degree markings along its edge.

The measure of a right angle is $90^\circ$.

Right angles are formed by perpendicular lines or segments.

The first step to construct an angle of $60^\circ$ using a compass and ruler is to draw a ray.

This ray will serve as one arm of the angle, with its endpoint being the vertex of the angle.

You can obtain an angle of $30^\circ$ from a $60^\circ$ angle using construction by bisecting the $60^\circ$ angle.

Bisecting an angle means dividing it into two equal angles.

The steps involved are:

1. First, construct a $60^\circ$ angle using a compass and ruler.

2. Place the compass point at the vertex of the $60^\circ$ angle.

3. Draw an arc that intersects both arms of the $60^\circ$ angle.

4. From each of the two points where the arc intersects the arms, draw an arc in the interior of the angle, using the same compass radius (the radius can be the same as or larger than the first arc's radius, as long as the two new arcs intersect). The compass point should be placed on the intersection points on the arms.

5. Draw a ray from the vertex of the angle through the point where these two new arcs intersect.

6. This new ray is the angle bisector, dividing the original $60^\circ$ angle into two equal angles, each measuring $\frac{60^\circ}{2} = 30^\circ$.

You can obtain an angle of $45^\circ$ from a $90^\circ$ angle using construction by bisecting the $90^\circ$ angle.

Bisecting an angle means dividing it into two equal angles.

The steps involved are:

1. First, construct a $90^\circ$ angle using a compass and ruler (e.g., by constructing a perpendicular to a line at a point on the line or by constructing a perpendicular bisector).

2. Place the compass point at the vertex of the $90^\circ$ angle.

3. Draw an arc that intersects both arms of the $90^\circ$ angle.

4. From each of the two points where the arc intersects the arms, draw an arc in the interior of the angle, using the same compass radius (or a larger radius, as long as the two new arcs intersect). The compass point should be placed on the intersection points on the arms.

5. Draw a ray from the vertex of the angle through the point where these two new arcs intersect.

6. This new ray is the angle bisector, dividing the original $90^\circ$ angle into two equal angles, each measuring $\frac{90^\circ}{2} = 45^\circ$.

The primary purpose of angle bisection is to divide a given angle into two equal angles.

This is a fundamental geometric construction used for several reasons, including:

1. To create new angles that are exactly half the measure of an existing angle (e.g., constructing $30^\circ$ from $60^\circ$, $45^\circ$ from $90^\circ$, $22.5^\circ$ from $45^\circ$, and so on).

2. To find the locus of points that are equidistant from the two arms of the angle.

In essence, it allows for precise division and manipulation of angles in geometric constructions.

Two specific angles that can be constructed easily using only a compass and ruler are:

1. An angle of $60^\circ$.

This is the basic angle formed by constructing an equilateral triangle using arcs of the same radius.

2. An angle of $90^\circ$ (a right angle).

This can be constructed by creating a perpendicular line, often using arcs to find points equidistant from two points on a line or from a point to a line.

In terms of practical geometry, the phrase "draw a line segment AB = 5 cm" means to construct a straight line that has a specific starting point and a specific ending point, and the distance between these two points is exactly 5 centimeters.

Here's what it involves practically:

1. Identifying a starting point: You begin by marking a point on your paper. Let's call this point A.

2. Using a ruler: You place a ruler on the paper, aligning the zero mark (or the starting edge depending on the ruler) precisely with the point A.

3. Measuring the length: While keeping the ruler steady with point A aligned with the zero mark, you look along the ruler to find the mark that corresponds to 5 centimeters. You then mark a second point at this 5 cm mark. Let's call this point B.

4. Drawing the segment: Finally, you draw a straight line connecting point A to point B along the edge of the ruler.

The result is a finite portion of a line (a line segment) whose endpoints are labeled A and B, and the length or distance between A and B is measured to be precisely 5 cm. The notation $AB = 5 \text{ cm}$ indicates that the length of the line segment with endpoints A and B is 5 centimeters.

When you draw a circle with a compass, the width you set the compass to is the radius of the circle.

Given that the compass is set to a width of $3$ cm, the radius of the circle is $r = 3$ cm.

The diameter of a circle ($D$) is twice its radius ($r$). The relationship is given by:

Substitute the value of the radius into the formula:

Therefore, the diameter of the circle is $6$ cm.

When constructing a perpendicular bisector, the two arcs drawn from each endpoint of the line segment must intersect at two points.

One intersection point is typically above the line segment, and the other is below it. The line connecting these two intersection points is the perpendicular bisector of the original line segment.

Yes, a perpendicular can be drawn from a point on a line to the same line. This is a standard construction in geometry.

Explanation:

Drawing a perpendicular from a point on a line to the same line essentially means constructing a line that passes through the given point and is at a $90^\circ$ angle to the original line at that point.

Here's the conceptual process (which corresponds to the construction using a compass and ruler):

1. Start with the given line and the specific point on that line.

2. Using the point as the center, draw arcs that intersect the line on both sides of the point. These two intersection points are equidistant from the original point.

3. From each of these two intersection points, draw arcs on the same side of the line (or on opposite sides, as long as they intersect) with a radius larger than the distance from the original point to the intersection points on the line.

4. These two new arcs will intersect at a point.

5. Draw a line connecting the original point on the line to this new intersection point.

This newly drawn line passes through the given point and forms a $90^\circ$ angle with the original line at that point. Therefore, it is a perpendicular to the line through the point on the line.

When bisecting an angle, the point of intersection of the two arcs drawn from the points on the angle's arms (using equal radii) is equidistant from both arms of the angle.

This property is crucial because any point that lies on the angle bisector is equidistant from the two lines (rays) that form the angle.

By connecting the vertex of the angle to this intersection point, you are drawing a line (or ray) that contains all points equidistant from the arms, thus creating the angle bisector, which divides the original angle into two equal parts.

In summary, the significance is that this point uniquely identifies a location that, when connected to the vertex, defines the line that perfectly splits the angle into halves based on the geometric property of angle bisectors.

If you need to draw a line segment of length $7.5$ cm and the ruler has one side marked in centimetres, you would use the side marked in centimetres.

This is because the required unit of measurement for the line segment's length is centimetres, and that side of the ruler provides the necessary markings in centimetres to accurately measure and draw the segment.

In practical geometry, the term 'construction' refers to the process of creating geometric figures using only a limited set of allowed tools, typically a compass and an unmarked straightedge (ruler).

Key aspects of construction:

1. Limited Tools: Only compass and straightedge are permitted. Measuring tools like protractors are generally not allowed unless explicitly stated for a specific type of construction.

2. Based on Principles: Constructions rely on fundamental geometric principles and postulates to accurately create figures like angle bisectors, perpendicular lines, circles, specific angles ($60^\circ$, $90^\circ$, etc.), and polygons.

3. Accuracy: The focus is on achieving geometric accuracy through the specific sequence of steps involving arcs (from the compass) and straight lines (from the straightedge), rather than relying on measurement alone.

In contrast, 'drawing' in geometry is a more general term that means representing geometric figures on paper or a surface. Drawing is less strict about the tools used and the method. You can use rulers for measuring, protractors for angles, set squares, or even freehand sketching.

The key difference is that construction emphasizes the method and the specific, limited tools used to achieve geometric accuracy based on principles, whereas drawing is about the representation of the figure, allowing for a wider range of tools and often involving measurement for approximation.

Yes, you can construct an angle of $75^\circ$ using only a compass and ruler.

Explanation:

An angle of $75^\circ$ can be obtained by combining constructible angles. Specifically, $75^\circ$ can be expressed as $60^\circ + 15^\circ$ or $90^\circ - 15^\circ$.

Since $60^\circ$ and $90^\circ$ are constructible angles using a compass and ruler, we can use them to construct $75^\circ$. The angle $15^\circ$ can be obtained by bisecting a $30^\circ$ angle ($30^\circ = 60^\circ/2$) or by bisecting the angle between a $90^\circ$ and a $60^\circ$ line drawn from the same vertex on the same base ray ($90^\circ - 60^\circ = 30^\circ$).

One common method for constructing $75^\circ$ is as follows:

1. Construct a $90^\circ$ angle.

2. Construct a $60^\circ$ angle sharing the same vertex and base ray as the $90^\circ$ angle, such that the $60^\circ$ angle is inside the $90^\circ$ angle.

3. The angle between the arm of the $60^\circ$ angle and the arm of the $90^\circ$ angle is $90^\circ - 60^\circ = 30^\circ$.

4. Bisect this $30^\circ$ angle using the compass.

5. The bisector divides the $30^\circ$ angle into two $15^\circ$ angles.

6. The angle formed by the base ray and the bisector ray is the sum of the $60^\circ$ angle and one of the $15^\circ$ angles, i.e., $60^\circ + 15^\circ = 75^\circ$.

Therefore, $75^\circ$ is a constructible angle using only a compass and ruler.

Steps to Construct a Circle with a Radius of $4$ cm:

1. Measure the radius: Take a ruler and a compass. Place the sharp point of the compass at the zero mark on the ruler.

2. Extend the pencil end of the compass until it reaches the $4$ cm mark on the ruler. This sets the opening of the compass to the desired radius of $4$ cm.

3. Mark the center: On your paper, mark a point where you want the center of the circle to be. Let's label this point as O.

4. Draw the circle: Place the sharp point of the compass firmly on the marked point O (the center). Hold the compass by its top and slowly rotate it around the point O, keeping the pencil end in contact with the paper.

5. Continue rotating the compass until the pencil end draws a complete circle.

Drawing and Labeling:

The drawn figure will be a circle. Label the point O at the center of the circle.

To label the radius, draw a line segment from the center O to any point on the circumference of the circle. Label this point on the circumference as A. The line segment OA is the radius. Write "$r = 4$ cm" next to the segment OA or near the circle.

Diameter of the Circle:

The diameter ($D$) of a circle is twice its radius ($r$).

Given the radius $r = 4$ cm.

The diameter of this circle is $8$ cm.

Steps to Construct a Line Segment of length $6.5$ cm using a Ruler:

1. Mark a starting point: Place your paper on a flat surface. Using a pencil and ruler, mark a point on the paper. Let's call this point A.

2. Align the ruler: Place the ruler such that the zero mark on the ruler's centimetre scale is exactly aligned with point A.

3. Mark the endpoint: Look along the edge of the ruler. Find the mark corresponding to $6.5$ cm. This mark will be exactly halfway between the 6 cm mark and the 7 cm mark. Mark a second point at this $6.5$ cm mark. Let's call this point B.

4. Draw the segment: Draw a straight line carefully along the edge of the ruler, connecting point A to point B.

You have now constructed the line segment AB with a length of $6.5$ cm.

Steps to Bisect Line Segment AB using a Compass and Ruler:

1. Set compass radius: Place the sharp point of the compass on point A (one endpoint of the segment). Open the compass to a radius that is more than half the length of AB. Since AB is 6.5 cm, the radius should be more than $6.5/2 = 3.25$ cm. For example, set it to about 4 cm or 5 cm.

2. Draw arcs from A: With the compass point at A and the radius set as above, draw an arc that extends both above and below the line segment AB.

3. Draw arcs from B: Without changing the compass radius, move the sharp point to point B (the other endpoint of the segment). Draw another arc that also extends above and below the line segment AB.

4. Identify intersection points: The two arcs drawn from A and B should intersect at two distinct points. Let's label the upper intersection point as P and the lower intersection point as Q.

5. Draw the bisector: Using the ruler, draw a straight line that connects point P to point Q.

6. Label the midpoint: The line PQ is the perpendicular bisector of the line segment AB. The point where the line PQ intersects the line segment AB is the midpoint of AB. Label this intersection point as M.

Point M is the midpoint of AB, so the length of segment AM is equal to the length of segment MB ($AM = MB = 6.5/2 = 3.25$ cm).

Drawing and Labeling:

Your drawing should show the line segment AB of length 6.5 cm.

It should show the arcs drawn from A and B intersecting at points P and Q above and below the segment.

It should show the line segment PQ drawn through P and Q, intersecting AB at point M.

Label the points A, B, P, Q, and M. Indicate the length AB = 6.5 cm and label M as the midpoint.

You can also add the symbol for perpendicularity at M where PQ intersects AB, indicating that the angle formed is $90^\circ$.

Steps to Construct a Perpendicular to a Line Through a Point On the Line:

1. Draw the line and point: Draw a straight line AB. Mark a point P anywhere on this line.

2. Draw initial arcs from P: Place the sharp point of the compass on point P. With any convenient radius (ensure the radius is small enough so the arcs stay on your paper, but large enough to clearly intersect the line), draw arcs that intersect the line AB on both sides of P. Let these intersection points be X and Y.

3. Set compass for intersecting arcs: Now, place the sharp point of the compass on point X. Open the compass to a radius that is greater than the distance PX (or PY). It's common and effective to use a radius slightly larger than the distance XY, or simply use the distance XY as the radius if space permits. Draw an arc above (or below) the line AB.

4. Draw second intersecting arc: Without changing the compass radius, move the sharp point to point Y. Draw another arc above (or below) the line AB, ensuring this arc intersects the arc drawn from X. Let the point of intersection of these two arcs be Q.

5. Draw the perpendicular line: Using the ruler, draw a straight line connecting point P to point Q.

The line PQ is perpendicular to the line AB and passes through the point P. The angle $\angle \text{APQ}$ and $\angle \text{BPQ}$ are both $90^\circ$.

Drawing and Showing Construction Marks:

Your drawing should show:

- The line AB with point P marked on it.

- Arcs from P intersecting AB at X and Y.

- Arcs from X and Y intersecting at Q (above or below AB).

- The line PQ drawn through P and Q.

Label points A, B, P, X, Y, and Q clearly.

Steps to Construct a Perpendicular from a Point Outside the Line to the Line:

1. Draw the line and point: Draw a straight line XY. Mark a point Q that is not on the line XY.

2. Draw initial arc from Q: Place the sharp point of the compass on point Q. Open the compass to a radius that is large enough to intersect the line XY at two distinct points. Draw an arc from Q that crosses the line XY at two points. Let these intersection points be A and B.

3. Set compass for intersecting arcs: Now, place the sharp point of the compass on point A. Open the compass to a radius that is greater than half the length of the segment AB. Draw an arc below the line XY (or on the side opposite to Q, if Q is above XY).

4. Draw second intersecting arc: Without changing the compass radius, move the sharp point to point B. Draw another arc below the line XY (or on the side opposite to Q) that intersects the arc drawn from A. Let the point of intersection of these two arcs be R.

5. Draw the perpendicular line: Using the ruler, draw a straight line connecting point Q to point R.

6. Identify the foot of the perpendicular: The line QR intersects the original line XY. The point where QR intersects XY is the foot of the perpendicular from Q to XY. Let's label this point as M.

The line QR is perpendicular to the line XY. The line segment QM is the perpendicular segment from the point Q to the line XY.

Drawing and Showing Construction Marks:

Your drawing should show:

- The line XY.

- Point Q located off the line XY.

- An arc centered at Q intersecting XY at points A and B.

- Arcs centered at A and B (with radius greater than AB/2) intersecting at point R (typically on the side of XY opposite to Q).

- The line QR drawn through Q and R.

- The point M where QR intersects XY.

Label the points X, Y, Q, A, B, R, and M clearly. You can also draw the segment QM as the perpendicular from Q to XY.

Steps to Construct an Angle of $60^\circ$ using a Compass and Ruler:

1. Draw a ray: Draw a straight line and mark a point on it. This point will be the vertex of the angle. Let's call the point A and the ray starting from A as AB.

2. Draw an arc: Place the sharp point of the compass on A. With any convenient radius, draw an arc that intersects the ray AB. Let the intersection point be C.

3. Draw a second arc: Without changing the compass radius, place the sharp point of the compass on point C. Draw another arc that intersects the first arc (drawn in step 2) at a point. Let this intersection point be D.

4. Draw the second arm: Using the ruler, draw a straight line (or ray) from point A through point D.

The angle $\angle \text{DAB}$ is the required $60^\circ$ angle.

Drawing and Labeling:

Your drawing should show:

- A ray AB with vertex A.

- An arc centered at A intersecting AB at C.

- An arc centered at C with the same radius, intersecting the first arc at D.

- A ray AD drawn from A through D.

Label points A, B, C, and D. Indicate the angle $\angle \text{DAB}$.

Explanation why this construction results in a $60^\circ$ angle:

In this construction, we have three points: A, C, and D.

By construction, the distance AC is the radius used in step 2. The arc from A ensures that all points on it are equidistant from A. So, AD = radius.

Also, by construction, the distance CD is the radius used in step 3, which is the same as the radius used in step 2. So, AC = CD = radius.

Furthermore, point C lies on the ray AB, and point D is the intersection of the two arcs.

Since AC = radius, AD = radius, and CD = radius, the triangle formed by connecting points A, C, and D (if we were to draw the segments AC and CD) would have all three sides equal in length (AC = AD = CD = radius).

A triangle with all three sides of equal length is called an equilateral triangle.

A property of equilateral triangles is that all three of their interior angles are equal. The sum of angles in any triangle is $180^\circ$.

In triangle ACD (if drawn), the angle at vertex A is $\angle \text{CAD}$. Since ACD is an equilateral triangle (based on our construction with equal radii), the angle $\angle \text{CAD}$ is $60^\circ$.

The ray AB lies along the segment AC, so $\angle \text{DAB}$ is the same as $\angle \text{CAD}$.

Thus, the angle $\angle \text{DAB}$ constructed this way measures exactly $60^\circ$.

Steps to Construct an Angle of $90^\circ$ using a Compass and Ruler (Perpendicular at a point on a line):

1. Draw a line and point: Draw a straight line. Mark a point P on this line where you want the vertex of the $90^\circ$ angle to be. Let the line be L.

2. Draw initial arcs from P: Place the sharp point of the compass on P. With any convenient radius, draw arcs that intersect the line L on both sides of P. Let the intersection points be A and B.

3. Set compass for intersecting arcs: Now, place the sharp point of the compass on point A. Open the compass to a radius that is greater than the distance AP (which is equal to BP). A radius slightly larger than AB (or significantly larger than AP/BP) works well. Draw an arc above the line L.

4. Draw second intersecting arc: Without changing the compass radius, move the sharp point to point B. Draw another arc above the line L that intersects the arc drawn from A. Let the point of intersection of these two arcs be Q.

5. Draw the perpendicular line: Using the ruler, draw a straight line connecting point P to point Q. Extend this line upwards.

The angle formed by the line PQ and the line L at point P is the required $90^\circ$ angle.

Drawing and Labeling:

Your drawing should show:

- The line L with point P on it.

- Arcs from P intersecting L at A and B.

- Arcs from A and B intersecting at Q (above L).

- The line PQ drawn through P and Q.

Label points P, A, B, and Q. Indicate the angle $\angle \text{APQ}$ (or $\angle \text{BPQ}$) as $90^\circ$ using the square symbol for a right angle.

Explanation of the geometry behind this construction:

In this construction, we have created points A and B on the line L such that P is the midpoint of the segment AB (because PA = PB, the radius used in step 2).

We then found a point Q such that QA = QB (because the same radius was used for arcs from A and B in steps 3 and 4).

Consider the points P, A, B, and Q. If we draw the line segment PQ, we have a line that passes through P. Points Q, A, and B form a triangle $\triangle \text{QAB}$. Point P lies on the base AB of this triangle.

The line segment PQ connects the vertex Q to the base AB, passing through P.

Since QA = QB, the triangle $\triangle \text{QAB}$ is an isosceles triangle with apex Q and base AB.

Since PA = PB, P is the midpoint of the base AB.

In an isosceles triangle, the line segment drawn from the apex to the midpoint of the base is perpendicular to the base.

Therefore, the line segment PQ is perpendicular to the base AB of $\triangle \text{QAB}$. Since A, P, and B lie on the line L, the line PQ is perpendicular to the line L at point P.

A perpendicular line forms a $90^\circ$ angle with the line it intersects. Thus, the angle $\angle \text{APQ}$ (or $\angle \text{BPQ}$) is $90^\circ$.

Steps to Bisect a Given Angle $\angle ABC$ using a Compass and Ruler:

1. Draw the angle: Draw the given angle $\angle \text{ABC}$ with vertex at B and arms BA and BC. For this problem, draw an obtuse angle, like one greater than $90^\circ$.

2. Draw an arc from the vertex: Place the sharp point of the compass on the vertex B. With any convenient radius, draw an arc that intersects both arms of the angle, BA and BC. Let the intersection point on arm BA be P and the intersection point on arm BC be Q.

3. Draw intersecting arcs from P and Q: Now, place the sharp point of the compass on point P. With a radius that is greater than half the distance between P and Q, draw an arc in the interior of $\angle \text{ABC}$.

4. Without changing the compass radius, move the sharp point to point Q. Draw another arc in the interior of $\angle \text{ABC}$ that intersects the arc drawn from P. Let the point of intersection of these two arcs be R.

5. Draw the angle bisector: Using the ruler, draw a straight line (or ray) from the vertex B through point R.

The ray BR is the angle bisector of $\angle \text{ABC}$. It divides $\angle \text{ABC}$ into two equal angles: $\angle \text{ABR}$ and $\angle \text{CBR}$.

Drawing and Showing Construction Marks:

Your drawing should show:

- The angle $\angle \text{ABC}$ (obtuse).

- An arc centered at B intersecting BA at P and BC at Q.

- Arcs centered at P and Q (with radius greater than PQ/2) intersecting at R in the interior of $\angle \text{ABC}$.

- The ray BR drawn from B through R.

Label points A, B, C, P, Q, and R. Indicate $\angle \text{ABC}$ and the bisector BR.

Verification of Equal Angles (Geometric Property):

The construction ensures that the resulting angles $\angle \text{ABR}$ and $\angle \text{CBR}$ are equal due to the properties of congruent triangles.

Consider the points B, P, Q, and R. By construction:

Therefore, the triangles $\triangle \text{BPR}$ and $\triangle \text{BQR}$ are congruent by the SSS (Side-Side-Side) congruence criterion.

Since the triangles are congruent, their corresponding parts are equal (CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

The angles $\angle \text{PBR}$ and $\angle \text{QBR}$ are corresponding angles in these congruent triangles.

The ray BR is positioned such that $\angle \text{PBR}$ is the same as $\angle \text{ABR}$, and $\angle \text{QBR}$ is the same as $\angle \text{CBR}$.

This geometric property confirms that the construction correctly divides the angle $\angle \text{ABC}$ into two equal angles, and the ray BR is indeed its angle bisector.

This problem involves three main construction steps:

1. Constructing the line segment PQ of length $7$ cm.

2. Constructing a perpendicular line at point P on PQ.

3. Constructing a $60^\circ$ angle at point Q with QP as one arm.

Steps for Construction:

Part 1: Construct Line Segment PQ = 7 cm

1. Draw a straight line and mark a point P on it.

2. Place the zero mark of the ruler at point P. Measure and mark a point Q at a distance of $7$ cm from P along the line.

3. Draw a line segment connecting P and Q.

Segment PQ is now constructed with length $7$ cm.

Part 2: Construct a Perpendicular at point P to PQ

1. Place the sharp point of the compass on P. With any convenient radius, draw arcs intersecting the line PQ on both sides of P. Let the intersection points be A and B.

2. Place the compass point on A. Open the compass to a radius greater than AP. Draw an arc above PQ.

3. Without changing the radius, place the compass point on B. Draw another arc above PQ, intersecting the previous arc. Let the intersection point be R.

4. Draw a straight line from P through R. Let's call this line L.

Line L is perpendicular to PQ at P.

Part 3: Construct an Angle of $60^\circ$ at point Q with arm QP

1. Place the sharp point of the compass on Q. With any convenient radius, draw an arc that intersects the ray QP. Let the intersection point be C.

2. Without changing the compass radius, place the sharp point on point C. Draw another arc that intersects the first arc (drawn from Q) at a point. Let this intersection point be D.

3. Draw a straight line (or ray) from Q through D. Let's call this ray QM.

The angle $\angle \text{PQM}$ (or $\angle \text{MQ P}$) is the required $60^\circ$ angle.

Drawing and Showing Construction Marks:

Your drawing should clearly show:

- The line segment PQ, labeled, with length 7 cm indicated.

- The arcs from P intersecting PQ at A and B.

- The arcs from A and B intersecting at R.

- The line PR extending upwards from P, showing the construction of the perpendicular.

- The arc from Q intersecting QP at C.

- The arc from C intersecting the first arc at D.

- The ray QM drawn from Q through D, showing the construction of the $60^\circ$ angle.

Label all relevant points (P, Q, A, B, R, C, D, M).

We can construct $30^\circ$ and $45^\circ$ angles by first constructing the basic $60^\circ$ and $90^\circ$ angles and then bisecting them, as bisecting an angle divides its measure exactly in half.

Construction of a $30^\circ$ Angle:

A $30^\circ$ angle is half of a $60^\circ$ angle ($60^\circ / 2 = 30^\circ$).

So, the steps are:

1. Construct a $60^\circ$ angle:

a. Draw a ray, say OA.

b. With O as the center and any convenient radius, draw an arc intersecting OA at point P.

c. With P as the center and the *same* radius, draw an arc intersecting the first arc at point Q.

d. Draw the ray OQ. The angle $\angle \text{AOQ}$ is $60^\circ$.

2. Bisect the $60^\circ$ angle:

a. With P as the center and a radius greater than half of PQ, draw an arc in the interior of $\angle \text{AOQ}$.

b. With Q as the center and the *same* radius, draw another arc intersecting the previous arc at point R.

c. Draw the ray OR.

The ray OR is the angle bisector of $\angle \text{AOQ}$. Therefore, $\angle \text{AOR} = \angle \text{QOR} = \frac{1}{2} \angle \text{AOQ} = \frac{1}{2} \times 60^\circ = 30^\circ$.

The angle $\angle \text{AOR}$ (or $\angle \text{QOR}$) is the required $30^\circ$ angle.

Construction of a $45^\circ$ Angle:

A $45^\circ$ angle is half of a $90^\circ$ angle ($90^\circ / 2 = 45^\circ$).

So, the steps are:

1. Construct a $90^\circ$ angle:

a. Draw a line, say XY, and mark a point O on it.

b. With O as the center and any convenient radius, draw arcs intersecting XY at points A and B (on either side of O).

c. With A as the center and a radius greater than AO, draw an arc above XY.

d. With B as the center and the *same* radius, draw another arc intersecting the previous arc at point C.

e. Draw the ray OC. The angle $\angle \text{XOC}$ (or $\angle \text{YOC}$) is $90^\circ$. Let's consider $\angle \text{YOC}$ for bisection, where OY is one arm and OC is the other.

2. Bisect the $90^\circ$ angle:

a. The arm OY intersects the initial arc from O at point B. The arm OC intersects the initial arc from O at point P (the same point C from the $90^\circ$ construction using arcs from A and B might land on the initial arc if the radius is large enough, or you might need to draw a new arc from O if the radius was small. Let's assume the ray OC intersects the initial arc from O at P for clarity of bisection steps, or simply use points from the $90^\circ$ construction that lie on the arms). Let's use the points where the initial arc from O intersects the arms OY and OC. Let the intersection on OY be B and the intersection on OC be P (using the point from the $90^\circ$ construction). Or, alternatively, with O as center, draw an arc that intersects OY at B and OC at P.

b. With B as the center and a radius greater than half of BP, draw an arc in the interior of $\angle \text{YOC}$.

c. With P as the center and the *same* radius, draw another arc intersecting the previous arc at point Q.

d. Draw the ray OQ.

The ray OQ is the angle bisector of $\angle \text{YOC}$. Therefore, $\angle \text{YOQ} = \angle \text{COQ} = \frac{1}{2} \angle \text{YOC} = \frac{1}{2} \times 90^\circ = 45^\circ$.

The angle $\angle \text{YOQ}$ (or $\angle \text{COQ}$) is the required $45^\circ$ angle.

Solution:

Steps for Construction:

1. Draw the Circle:

a. Take a ruler and a compass. Set the compass opening to $3.5$ cm by placing the sharp point at the 0 mark and extending the pencil lead to the $3.5$ cm mark.

b. Mark a point O on your paper; this will be the center of the circle.

c. Place the sharp point of the compass on O and draw a circle with the set radius.

2. Draw a Chord AB:

a. Mark any two points A and B on the circumference of the circle.

b. Using a ruler, draw a straight line segment connecting points A and B. This segment AB is a chord of the circle.

3. Construct the Perpendicular Bisector of Chord AB:

a. Place the sharp point of the compass on point A (one endpoint of the chord). Open the compass to a radius that is greater than half the length of the chord AB. (Estimate visually or measure roughly).

b. With this radius and center A, draw an arc that extends above and below the chord AB.

c. Without changing the compass radius, place the sharp point on point B (the other endpoint of the chord). Draw another arc that extends above and below the chord AB, ensuring it intersects the first arc in two distinct points.

d. Let the two points where the arcs intersect be P and Q.

e. Using a ruler, draw a straight line connecting points P and Q.

The line PQ is the perpendicular bisector of the chord AB.

Drawing and Observation:

Draw the circle, the center O, the chord AB, and the construction arcs from A and B intersecting at P and Q. Then draw the line PQ.

Observe the line PQ. You will notice that the line PQ passes through the center O of the circle.

Conclusion from Observation:

Yes, the perpendicular bisector of the chord AB passes through the centre O of the circle.

Geometric Property:

The geometric property related to the perpendicular bisector of a chord in a circle is:

"The perpendicular bisector of any chord of a circle always passes through the center of the circle."

This property is true for all chords in a circle and is a fundamental theorem in geometry.

Given: A line L and a point M on line L.

To Construct: An angle of $120^\circ$ at point M on line L and then bisect this angle.

Construction Steps:

Part 1: Constructing the $120^\circ$ Angle

1. Draw a straight line L and mark a point M anywhere on this line.

2. Place the sharp point of the compass on point M. With any convenient radius, draw an arc that intersects the line L at two points. Let the point to the left of M be A and the point to the right of M be B.

3. Place the sharp point of the compass on point B. With the *same* radius used in step 2, draw an arc that intersects the first arc (the one centered at M) at a point. Let this point be C.

4. Using a ruler, draw a straight line (or ray) from point M through point C.

The angle $\angle \text{AMC}$ is the constructed $120^\circ$ angle.

Part 2: Bisecting the $120^\circ$ Angle $\angle$AMC

1. The arc drawn in Part 1, step 2, from M intersects the arm MA of the angle $\angle \text{AMC}$ at point A, and the arm MC at point C. (Note: These are the same points A and C used in the $120^\circ$ construction).

2. Place the sharp point of the compass on point A. With a radius that is greater than half the distance between A and C, draw an arc in the interior of $\angle \text{AMC}$.

3. Without changing the compass radius, move the sharp point to point C. Draw another arc in the interior of $\angle \text{AMC}$ that intersects the arc drawn from A. Let the point of intersection of these two arcs be D.

4. Using a ruler, draw a straight line (or ray) from the vertex M through point D.

The ray MD is the angle bisector of $\angle \text{AMC}$.

Measure of the Resulting Angles:

When the $120^\circ$ angle $\angle \text{AMC}$ is bisected by the ray MD, it means the angle is divided into two equal angles.

The measure of each of these smaller angles is half the measure of the original angle $\angle \text{AMC}$.

The measure of each of these smaller angles is $60^\circ$.

Given: A line segment of length $8$ cm.

To Divide: The line segment into four equal parts using compass and ruler constructions.

Explanation of the Method:

To divide a line segment into four equal parts, we can use the construction of a perpendicular bisector multiple times. Bisecting a segment divides it into two equal halves. If we bisect the original segment, we get two equal parts. If we then bisect each of these two parts, we will have divided the original segment into $2 \times 2 = 4$ equal parts.

Construction Steps:

Step 1: Construct the line segment AB of length 8 cm.

1. Draw a straight line and mark a point A on it.

2. Using a ruler, measure $8$ cm from point A along the line and mark point B. Draw the line segment AB.

Step 2: Bisect the line segment AB to find its midpoint (First Bisection).

1. Place the sharp point of the compass on point A. Open the compass to a radius that is greater than half the length of AB (i.e., $> 4$ cm). Draw arcs above and below the line segment AB.

2. Without changing the compass radius, place the sharp point on point B. Draw arcs above and below the line segment AB, ensuring they intersect the arcs drawn from A. Let the intersection points be P and Q.

3. Using a ruler, draw a straight line connecting point P to point Q. This line is the perpendicular bisector of AB. Let the point where PQ intersects AB be M$\mathbf{_1}$.

Point M$\mathbf{_1}$ is the midpoint of AB, so AM$\mathbf{_1}$ = M$\mathbf{_1}$B = $8/2 = 4$ cm.

Step 3: Bisect the line segment AM$\mathbf{_1}$ to find its midpoint (Second Bisection).

1. Place the sharp point of the compass on point A. Open the compass to a radius that is greater than half the length of AM$\mathbf{_1}$ (i.e., $> 4/2 = 2$ cm). Draw arcs above and below the line segment AM$\mathbf{_1}$.

2. Without changing the compass radius, place the sharp point on point M$\mathbf{_1}$. Draw arcs above and below the line segment AM$\mathbf{_1}$, ensuring they intersect the arcs drawn from A. Let the intersection points be R and S.

3. Using a ruler, draw a straight line connecting point R to point S. This line is the perpendicular bisector of AM$\mathbf{_1}$. Let the point where RS intersects AM$\mathbf{_1}$ be M$\mathbf{_2}$.

Point M$\mathbf{_2}$ is the midpoint of AM$\mathbf{_1}$, so AM$\mathbf{_2}$ = M$\mathbf{_2}$M$\mathbf{_1}$ = $4/2 = 2$ cm.

Step 4: Bisect the line segment M$\mathbf{_1}$B to find its midpoint (Third Bisection).

1. Place the sharp point of the compass on point M$\mathbf{_1}$. Open the compass to a radius that is greater than half the length of M$\mathbf{_1}$B (i.e., $> 4/2 = 2$ cm). Draw arcs above and below the line segment M$\mathbf{_1}$B.

2. Without changing the compass radius, place the sharp point on point B. Draw arcs above and below the line segment M$\mathbf{_1}$B, ensuring they intersect the arcs drawn from M$\mathbf{_1}$. Let the intersection points be T and U.

3. Using a ruler, draw a straight line connecting point T to point U. This line is the perpendicular bisector of M$\mathbf{_1}$B. Let the point where TU intersects M$\mathbf{_1}$B be M$\mathbf{_3}$.

Point M$\mathbf{_3}$ is the midpoint of M$\mathbf{_1}$B, so M$\mathbf{_1}$M$\mathbf{_3}$ = M$\mathbf{_3}$B = $4/2 = 2$ cm.

Result:

The line segment AB of length $8$ cm is now divided into four equal parts by the points M$\mathbf{_2}$, M$\mathbf{_1}$, and M$\mathbf{_3}$.

The four equal parts are AM$\mathbf{_2}$, M$\mathbf{_2}$M$\mathbf{_1}$, M$\mathbf{_1}$M$\mathbf{_3}$, and M$\mathbf{_3}$B.

The length of each part is $2$ cm, since $AM_2 = M_2M_1 = M_1M_3 = M_3B = 2$ cm, and $2+2+2+2=8$ cm.

Drawing and Construction Marks:

Your drawing should show the line segment AB, the points M$\mathbf{_1}$, M$\mathbf{_2}$, and M$\mathbf{_3}$ clearly marked on it, and all the construction arcs used for the three bisections (arcs from A & B, arcs from A & M$\mathbf{_1}$, and arcs from M$\mathbf{_1}$ & B).