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