two tangents PA and PB are drawn to a circle with center O from an external point P. then prove that-

a) angle APB = 2 angle OAB

b) if OP = 10 cm, AP = 8cm. find OA=?

 given : a circle with centre O.

PA and PB are two tangents to the circle at A and B

To prove : L APB = 2 L OAB

Proof : join AB and OA

OA is perpendicular to PA (tangent and the radius at the point of contact)

therefore LOAP = 90degree

let L OAB = x

therefore L PAB = 90 - x ----- (1)

PA = PB (tangent from same external point)

therefore triangle APB is isosceles

therefore LPAB = L PBA = 90-x (angles opp to equal sides)

Tn triangle APB 

LP + 90 - x + 90 - x = 180 degree (angle sum property)

LP + 180 - 2x = 180degree

LP = 2x

L APB = 2x

ie, L APB = 2 L OAB

 b

angle PAO = 90 

Therefore

By pythagoras

PO² = PA² + AO²

(10)² - (8)² =AO² 

100- 64 =AO²

36 =AO²

6 = AO