Question

in this figure AB is the diameter of the circle with centre O. AQ, BP and PRQ are tangents. Prove that OP and OQ are perpendicular to each other.

Answer 1

Given that AQ, BP, and QP are tangents, we know that angles QAO, QRO, PBO, and PRO are right angles.

We know that OA, OB, and OR are radii and are therefore equal.

We can now use Pythagorean theorem to show that AQ = RQ, and BP = RP

Thus, by the SSS postulate, triangle AQO is congruent to triangle RQO, and triangle BPO is congruent to triangle RPO.

This now gives us angle AOQ = angle ROQ, and angle BOP = angle ROP.

Since AOQ + ROQ + BOP + ROP = 180

We have ROQ + ROQ + ROP + ROP = 180

2ROQ + 2ROP = 180

ROQ + ROP = 90

POQ = 90

And OP and OQ are perpendicular.