A circle with centre at O is touching two intersecting lines AX and BY The two points of contact A a...........

Quantitative Aptitude ( CCS) Geometry

A circle (with centre at O) is touching two intersecting lines AX and BY. The two points of contact A and B subtend an angle of 65° at any point C on the circumference of the circle. If P is the point of intersection of the two lines, then the measure of ∠APO is

25°

65°

90°

40°

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Explanation:

Given, ∠ACB = 65°
Note : The angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of the circle.
∴ ∠AOB = 2 × 65° = 130°
Note : A tangent at any point of a circle is perpendicular to the radius through the point of contact.
∴ ∠OAP = 90°
∠AOP = 65° [? ∠AOP = ∠AOB/2]
We know that, the sum of the three angles of a triangle is 180°.
∴ ∠APO = 180° – 90° – 65° = 25°