Question
Two tangents are drawn from an external point P to a circle with center O and radius r. If the distance between P and O is 2r, find the angle subtended by the tangents at P.
Solution
The tangents from an external point are equal in length. Let the angle subtended by the tangents at the center be 2θ, then OP is the hypotenuse of the right triangle OAP, where OA is the radius, and PA is the tangent. Using the relationship cos(θ) = r / 2r = 1/2, we get θ = 60°. Hence, the angle subtended by the tangents at P is 2 * 60° = 120°.
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