two tangents PA and PB are drawn to the circle with Centre O such that angle APB is equal to 120 degree prove that op =
2AP
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Answer:
Given: O is the centre of the circle. PA and PB are tangents drawn to a circle and ∠APB = 120°.
To prove: OP = 2AP
Proof:
In ΔOAP and ΔOBP,
OP = OP (Common)
∠OAP = ∠OBP (90°) (Radius is perpendicular to the tangent at the point of contact)
OA = OB (Radius of the circle)
∴ ΔOAP is congruent to ΔOBP (RHS criterion)
∠OPA = ∠OPB = 120°/2 = 60° (CPCT)
In ΔOAP,
cos∠OPA = cos 60° = AP/OP
Therefore, 1/2 =AP/OP
Thus, OP = 2AP
Hence, proved.
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