Q. 25.0( 4 Votes )

# A, B are the points on ⨀ (O, r) such that tangents at A and B intersect in P. Prove that OP is the bisector of ∠AOB and PO is the bisector of ∠APB.

Given that A and B are the points on (O, r) such that tangents at A and B intersect in P.

We have to prove that OP is the bisector of AOB and PO is the bisector of APB.

Proof:

In circle (O, r), AP is a tangent at A and BP is the tangent at B.

OAP = OBP = 90°

Considering ΔOAP and ΔOBP,

OA = OB (radius)

OP = OP (common segment)

By RHS theorem, ΔOAP = ΔOBP i.e. OAP and OBP is a congruence.

APO = BOP and AOP = BOP

Here, O is in the interior part of APB and P is in the interior part of AOB.

OP is the bisector of AOB and PO is the bisector of APB.

Hence proved.

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