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.

Answer :

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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