Find angle x x

Geometry Level 3

Two tangents to a circle with center O O meet at A A and extent an angle of 5 0 ∘ 50^\circ . Another tangent to the circle intersects the two tangents at B B and C C . If ∠ B O C = x ∘ \angle BOC = x^\circ , find x x .


The answer is 65.

Aly Ahmed by Aly Ahmed , 34, Egypt

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1 solution

We use two properties of a circle :

(i) Tangent is perpendicular to radius at the point of tangency.

(ii) from an external point the two tangents drawn to a circle have equal length upto the point of tangency.

Using these two properties, we get 25 ° + x 2 = 90 ° − x 2 25\degree+\dfrac{x}{2}=90\degree-\dfrac{x}{2} or x = 65 ° x=\boxed {65\degree}

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