AB is a chord of length 24 cm of a circle of radius 15 cm. The tangents at A and B intersect at a point P. Find the length PA.

 AB is a chord of length 24 cm of a circle of radius 15 cm. The tangents at A and B intersect at a point P. Find the length PA.

Let O be the centre of the circle and Q the midpoint of chord /AB

Given: OA = OB = 15 cm, AB = 24 cm

Q is midpoint of AB

AQ = QB = 24/2 = 12 cm,

also OQ ⊥ AB  

In right triangle  ΔOQA:

OA² = OQ² + AQ²

15² = OQ² + 12²

225 = OQ² + 144

OQ² = 81 

OQ = 9 cm

  ∠OAP = 90° since radius ⊥ tangent

so ΔOAQ ~ ΔOPA

using corresponding sides

OA/OP = OQ/OA

OA² = OQ × OP

15² = 9 × OP

225 = 9 × OP 

 OP = 25 cm 

In right triangle  ΔOPA:

 PA² = OP² − OA²

PA² = 25² − 15² = 625 − 225 = 400

PA = 20 cm  

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circles tangents to circles

a detailed video on an alternative method to solve the problem