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  • A is a point at a distance 13 cm from the centre O of a circle of radius 5 cm.AP and AQ are the tangents to the circle at P and Q. If a tangent BC is drawn at a point R lying on the minor arc PQ to intersect AP at B and AQ at C, find the perimeter of the

A is a point at a distance 13 cm from the centre O of a circle of radius 5 cm.AP and AQ are the tangents to the circle at P and Q. If a tangent BC is drawn at a point R lying on the minor arc PQ to intersect AP at B and AQ at C, find the perimeter of the \bigtriangleupABC.

Answers (1)

Answer 24 cm
Solution
Let us make figure according to question

 

Given : OA = 13 cm, Radius = 5 cm
Here AP = AQ            (tangent from same point)
OP \perp PA, OQ \perp QA  (  AP, AQ are tangents)
In \bigtriangleupOPA using Pythagoras theorem
\left ( H \right )^{2}= \left ( B \right )^{2}+\left ( P \right )^{2}
\left ( QP \right )^{2}= \left ( OP \right )^{2}+\left ( PA \right )^{2}
\left ( 13 \right )^{2}= \left ( 5 \right )^{2}+\left ( PA \right )^{2}
\left ( PA \right )^{2}= 169-25
PA= \sqrt{144}= 12\, cm ........(i)
Perimeter of \bigtriangleupABC = AB + BC + CA
 = AB + BR + RC + CA
= AB + BD + CQ + CA
  [ BP and BR are tangents from point B and CP and CQ are tangents from point C]
= AP + AQ      [  AP = AB + BP, AQ = AC + CQ]
 = AP + AP      [  AP = AQ]
 = 2AP
= 2 × 12           [using (i)]
= 24 cm


 

Posted by

infoexpert27

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