Question

if PA and PB all two tangents drawn to the circle with centre O such that angleBPA=120 degree,prove that OP=2PB.

Answer 1

In Δ APO and Δ BPO ;        OP=OP[common]   ;                 AP=BP[tangent]   ;         angle PAO=angle PBO                                                                                             by   SAS rule,                     ΔAPO congruent to Δ BPO    ;  angle APO= angle BPO =1/2 angle APB =1/2 × 120  =60 ; In   Δ BPO  BP/OP= Cos60³    ;  BP/OP= 1/2               SO, we get  OP=2PB     

Answer 2

n Δ APO and Δ BPO ;        OP=OP[common]   ;                 AP=BP[tangent]   ;         angle PAO=angle PBO                                                                                             by   SAS rule,                     ΔAPO congruent to Δ BPO    ;  angle APO= angle BPO =1/2 angle APB =1/2 × 120  =60 ; In   Δ BPO  BP/OP= Cos60³    ;  BP/OP= 1/2               SO, we get  OP=2PB