Question

A line segment AB is bisected at point P and through point P another line segment PQ, which is perpendicular to AB, is drawn. Show that: QA = QB.

Answer 1

Given: A ΔABQ in which AB is bisected at P 
PQ is perpendicular to AB

We need to prove that

QA = QB

Proof:

In ΔAPQ and ΔBPQ

AP = PB                    ...[ P is the mid-point of AB ]

∠QPA = ∠QPB = 90°  ...[ PQ is perpendicular to AB ]

PQ = PQ                     ...[ Common] 

∴ By Side-Angel-Side criterion of congruence,

ΔQAP ≅ ΔQBP

The corresponding parts of the congruent triangles are

congruent.

∴ QA = QB   ...[ c.p.c.t ]