Question

In the figure below, AP and BQ are equal and parallel.

Prove that AM = MB. Can you suggest a construction to locate the midpoint of a line, using this idea?

Answer 1

Given, AP = BQ, AP || BQ

In ΔAMP and ΔBMQ,

∠APM = ∠MQB (Alternate interior angles)

AP = BQ (Given)

∠MAP = ∠MBQ (Alternate interior angles)

As one side and two angles of a triangle are equal to the one side and two angles of the other triangle, ΔAMP ≅ ΔBMQ.

Corresponding parts of congruent triangles are also congruent.

∴ AM = MB

To locate the midpoint of a line, we can follow the following steps of construction:

1) Draw a line segment AB of any desired length.

2) Draw an acute angle at point A on the upper side of AB.

3) Mark a point P on ray AT.

4) From point B, draw a ray BY parallel to ray AP on the lower side of AB.

5) Mark a point Q on AY such that AP = BQ.

6) Join PQ and mark its point of intersection with AB as M.

M is the midpoint of line segment AB.