Question

AP and PQ are tangents drawn from a point A to a circle with centre O and radius 9 cm. If OA = 15 cm, then AP + AQ = 

A. 12 cm 

B. 18 cm 

C. 24cm 

D. 36 cm

Answer 1

Answer  is C. 24cm

Given: 

Radius = 9 cm 

OA = 15 cm

Property 1: If two tangents are drawn to a circle from one external point, then their tangent segments (lines joining the external point and the points of tangency on circle) are equal. 

By the above property, 

AP = AQ (tangent from A) 

Property 2: The tangent at a point on a circle is at right angles to the radius obtained by joining center and the point of tangency. 

By above property, ∆POA is right-angled at ∠OAP (i.e., ∠OPA = 90°). 

Therefore by Pythagoras theorem, 

AP² + PO² = AO² 

⇒ AP² = AO² – PO² 

⇒ AP² = 15² – 9² 

⇒ AP² = 225 – 81 

⇒ AP2 = 144 

⇒ AP = √144 

⇒ AP = 12 

AP + AQ = 12 cm + 12 cm = 24 cm 

Hence, AP + AQ = 24 cm