Question

The given figure shows a circle with center O. P is mid-point of chord AB.

Show that OP is perpendicular to AB.

Answer 1

Given: in the figure, O is center of the circle, and AB is chord. P is a point on AB such that AP = PB.

We need to prove that, OP ⊥ AB

Construction: Join OA and OB

Proof:  

In ΔOAP and ΔOBP

OA = OB        ...[Radii of the same circle]

OP = OP        ...[Common]

AP = PB         ...[Given]

∴ By Side-Side-Side criterion of congruency,

ΔOAP ≅  ΔOBP 

The corresponding parts of the congruent triangles are congruent.

∴ ∠OPA = ∠OPB                     ...[By c.p.c.t]

But ∠OPA + ∠OPB = 180°       ...[Linear pair]

∴ ∠OPA = ∠OPB = 90°

Hence, OP ⊥ AB.