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Geometrical Constructions

    Introduction

    Basic Constructions

    Construction of Triangles (Special Cases)

    What We Have Discussed

    Easy Level Worksheet

    Moderate Level Worksheet

    Hard Level Worksheet

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Glossary

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Geometrical Constructions > Basic Constructions

Basic Constructions

Perpendicular Bisector of a segment

Steps for Construction:

  • Draw the given line segment. Let’s say the segment is AB.

  • Place the compass at point A. Set the compass width more than half of AB.

  • Draw an arc above and below the line.

  • Place the compass at point B

  • Keep the same compass width.

  • Draw another arc above and below the line, intersecting the previous arcs.

  • Mark the intersection points of the arcs

  • Label these intersection points as P and Q.

  • Draw a straight line through P and Q

This is the perpendicular bisector of AB i.e. it cuts AB exactly in the middle at point M and forms a 90° angle.

Why Does This Method Work?

Given: A line segment AB, and we construct a perpendicular bisector using arcs.

To Prove: The bisector divides AB into two equal parts and is perpendicular to it.

Proof:

The arcs drawn from A and B have the same radius, meaning: AP = and AQ =

So, ΔAPQ ≅ ΔBPQ (By congruence).

Since the two triangles are congruent, it follows that: AM =

So, M is the of AB.

Since ΔAPQ ≅ ΔBPQ, the angles at M are and sum to °.

Since both angles are equal, each is °, meaning PQ is to AB.

Bisector of an angle

Steps for Construction:

  • Draw the given angle. Let’s say the angle is ∠ABC, where B is the vertex.

  • Place the compass at B (the vertex).

  • Draw an arc that cuts both arms BA and BC at points P and Q.

  • Place the compass at P.

  • Using the same radius, draw an arc inside the angle.

  • Place the compass at Q.

  • Using the same radius, draw another arc inside the angle.

  • Let these two arcs intersect at point R.

  • Draw a straight line from B to R.

This is the angle bisector, dividing ∠ABC into two equal parts.

Why Does This Method Work?

Given: An angle ∠ABC, and we construct its bisector using arcs.

To Prove: The bisector divides ∠ABC into two equal angles.

Proof: Since BP = , the arcs from P and Q are equidistant from B.

In ΔBPR and ΔBQR:

BP = (same arc )

BR = ( side)

PR = (same compass setting)

So, ΔBPR ≅ ΔBQR by Congruence.

Since corresponding parts of congruent triangles are equal, ∠PBR = ∠

This means BR bisects ∠ABC into two equal halves.

Drawing 60 degrees

Steps for Construction:

Draw a ray

Let’s say we draw a ray AB, where A is the starting point. Place the compass at A

Set a convenient radius and draw an arc cutting the ray at point P. Keep the same compass width

Place the compass at P and draw another arc intersecting the first arc at point Q. Draw a line from A to Q

AQ is the required 60° angle with AB.

Why Does This Method Work?

Given: A ray AB, and we construct a 60° angle using a compass.

To Prove: ∠BAQ = 60°.

Proof: Equal Arcs with the Same Radius

Since the same compass radius was used, AP = PQ = QA, forming an triangle ΔAQP.

In ΔAQP, all sides are , so all angles are °.

∠BAQ = 60° (because it is one angle of the equilateral triangle).