Two tangent segments PA and PB are drawn to a circle with centre O such that angle APB is 120 degree. Prove that OP = 2 AP.
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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 )
∴ ΔOAP is congruent to ΔOBP (RHS )
∠OPA = ∠OPB = 120°/2 = 60° (CPCT)
In ΔOAP,
cos∠OPA = cos 60° = AP/OP
Therefore, 1/2 =AP/OP
Thus, OP = 2AP
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 )
∴ ΔOAP is congruent to ΔOBP (RHS )
∠OPA = ∠OPB = 120°/2 = 60° (CPCT)
In ΔOAP,
cos∠OPA = cos 60° = AP/OP
Therefore, 1/2 =AP/OP
Thus, OP = 2AP
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