Question

Two tangent PA and PB are drawn to the circle at the centre O such that angle APB=120 degree. Prove that OP= 2AP

Answer 1

GIVEN : O is the centre of the circle . PA and PB are tangents to the circle. Angle APB = 120
TO PROVE: OP= 2AP
PROOF:
In triangle OAP and triangle OBP,
OP = OP                             ( common)
angle OAP = Angle OBP ( radius is perpendicular to the tangent of the circle)
OA = OB                            ( radii of same circle)

theerfore, triangle OAP is congruent to triangle OBP       
 ( RHS congruence criterion)

angle OAP = angle OBP  ( CPCT)
Angle OAP = Angle OBP = 120/2 = 60.

In triangle OAP,
cos OPA = AP/OP
cos 60 = AP/OP
1/2= AP/OP
OP= 2AP.
HENCE PROVED.