Question 1

What are the two basic tools used in geometric constructions, and what are their primary functions?

Accepted Answer

Compass for drawing arcs, ruler for drawing straight lines without measurement markings — The two basic tools used in geometric constructions are the compass and the ruler (straightedge). The compass is used to draw arcs and circles, while the ruler is used to draw straight lines but without any measurement markings to ensure constructions are done geometrically, not by measurement.

Question 2

Why is it important to use only a compass and an unmarked ruler for geometric constructions instead of measuring tools like a protractor or a marked ruler?

Accepted Answer

Geometric constructions rely on the exactness of shapes and angles created through logical steps rather than measurements. Using only a compass and an unmarked ruler ensures that the constructions are precise and based on geometric principles, not on approximations from measurements. This method allows for creating figures that satisfy specific conditions exactly, such as perpendicular bisectors and angle bisectors. — Geometric constructions are designed to produce exact figures by following specific steps using only a compass and an unmarked ruler. Measuring tools like protractors or marked rulers introduce approximations and errors. The compass and unmarked ruler ensure that the properties of the figures, like equal lengths or right angles, are maintained through congruence and geometric reasoning rather than measurement.

Question 3

Describe the step-by-step procedure to construct the perpendicular bisector of a given line segment AB using a compass and an unmarked ruler.

Accepted Answer

To construct the perpendicular bisector of line segment AB: 
1. Place the compass pointer at point A and draw arcs above and below the line segment with a radius more than half of AB.
2. Without changing the compass width, draw similar arcs from point B, intersecting the previous arcs above and below AB.
3. Mark the points of intersection of the arcs as points C and D.
4. Using the ruler, draw a straight line through points C and D. This line is the perpendicular bisector of AB, dividing it into two equal parts at a right angle. — The construction works because points C and D are equidistant from A and B. Thus, the line joining C and D is perpendicular to AB and bisects it. This method does not require measuring the length or angle but uses the properties of congruent triangles formed by the arcs.

Question 4

In the construction of the perpendicular bisector of a line segment XY, why does the line joining the intersection points of arcs above and below XY pass through the midpoint of XY and form a right angle with it?

Accepted Answer

The line joining the intersection points passes through the midpoint of XY because these points are equidistant from X and Y. By proving that triangles formed by these points and X and Y are congruent using the SAS condition, it follows that the line bisects XY. The angles formed where this line meets XY are right angles because the sum of the angles at the intersection is 180°, and the congruence shows each is 90°, making the line perpendicular to XY. — Using congruence of triangles AOX and AOY, where AO is common and AX = AY, we find that the angles at O are equal and together form a straight line. This implies each angle is 90°, so the line is perpendicular and bisects XY at O, the midpoint.

Question 5

Fill in the blank: In geometric constructions, a line that divides a given line segment into two equal parts and is perpendicular to it is called a _____ bisector.

Accepted Answer

perpendicular — A perpendicular bisector divides a line segment into two equal parts at 90° angle, which is fundamental in many geometric constructions.

Question 6

True or False: Any point that is equidistant from the endpoints of a line segment lies on the perpendicular bisector of that segment.

Accepted Answer

True — By definition, the perpendicular bisector consists of all points equidistant from the endpoints of the segment. Hence, any such point lies on the perpendicular bisector.

Question 7

Match the following terms with their definitions:

1. Perpendicular bisector
2. Angle bisector
3. Compass
4. Straightedge

Accepted Answer

— Perpendicular bisector divides a line segment into two equal parts at right angles.
Angle bisector divides an angle into two equal parts.
Compass is a tool used to draw arcs and circles.
Straightedge is a ruler without markings used to draw straight lines.

Question 8

Explain how to construct a perpendicular to a given line l from a point P lying on the line using a compass and straightedge.

Accepted Answer

To construct a perpendicular to line l from point P on it: 
1. Place the compass pointer at P and draw arcs intersecting line l at two points A and B on either side of P.
2. With the compass pointer at A and radius more than half the distance AB, draw an arc above the line.
3. Repeat the same from point B with the same radius, creating an intersection point C above the line.
4. Draw a straight line from P through point C. This line is perpendicular to l at P. — The construction works because triangles PAC and PBC are congruent, ensuring the angle at P is 90°. This method does not require measuring angles but relies on congruent arcs and points equidistant from A and B.

Question 9

True or False: When constructing a perpendicular from a point outside a line, the intersection point of arcs drawn from points A and B on the line lies on the perpendicular from the external point.

Accepted Answer

True — The intersection point of arcs drawn from A and B is equidistant from A and B, ensuring that the line joining this point and the external point is perpendicular to the line.

Question 10

Fill in the blank: To construct the bisector of an angle ∠AOB, first draw an arc from the vertex O that intersects the arms of the angle at points _____ and _____.

Accepted Answer

C / D — The arc from vertex O intersects the arms of the angle at points C and D, which are then used to draw arcs that intersect at point E, through which the bisector is drawn.

Question 11

Describe the steps to construct the bisector of a given angle ∠AOB using a compass and straightedge.

Accepted Answer

To bisect angle ∠AOB: 
1. Place the compass pointer at vertex O and draw an arc intersecting both arms of the angle at points C and D.
2. Without changing the compass width, draw arcs from points C and D such that these arcs intersect at point E inside the angle.
3. Draw a straight line from vertex O through point E. This line OE is the angle bisector, dividing the angle into two equal parts. — The construction ensures that point E is equidistant from the arms of the angle, making OE the bisector. This method does not require measuring the angle but uses congruent arcs and intersection points.

Question 12

Why does the line drawn from vertex O through the intersection point E of arcs from points C and D bisect the angle ∠AOB?

Accepted Answer

The point E is equidistant from the arms OC and OD because it lies at the intersection of arcs drawn from points C and D with equal radii. Therefore, the line OE divides the angle ∠AOB into two equal angles, making OE the angle bisector. — Since E is equidistant from both arms of the angle, the triangles formed on either side of OE are congruent, ensuring that ∠BOE = ∠AOE, thus bisecting the angle.

Question 13

Construct a 45° angle using only a compass and straightedge. Which construction steps would you use?

Accepted Answer

First, construct a 90° angle using the perpendicular construction method. Then, bisect this 90° angle using the angle bisector construction steps to obtain a 45° angle. — Since 45° is half of 90°, bisecting a right angle using the angle bisector method yields a 45° angle. This method uses only compass and straightedge without measuring the angle.

Question 14

True or False: The angle bisector of an angle has the property that any point on it is equidistant from the sides of the angle.

Accepted Answer

True — By definition, the angle bisector divides the angle into two equal parts, and any point on the bisector is equidistant from the two arms (sides) of the angle.

Question 15

Explain the method to copy a given angle ∠ABC to a different point using only a compass and straightedge.

Accepted Answer

To copy angle ∠ABC: 
1. Draw a ray AX where the copied angle will be constructed.
2. With the compass pointer at vertex B, draw an arc that intersects the arms of the angle at points C and D.
3. Using the same radius, draw an arc from point A intersecting ray AX at point Z.
4. Measure the length CD using the compass.
5. With the compass pointer at Z, mark point Y on the arc such that YZ = CD.
6. Draw ray AY. Then ∠XAY is congruent to ∠ABC. — By constructing congruent arcs and transferring the chord length, the triangles formed are congruent by SSS, ensuring the copied angle equals the original angle.

Constructions

Constructions — Study Notes

Introduction to Geometric Constructions

Constructing a Perpendicular Bisector of a Line Segment

Constructing a Perpendicular to a Line from a Point on the Line

Practice Questions — Constructions

All 7 Chapters in Ganita Prakash-II