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AB is a chord of a circle in minor segment with center O. C is a point on the minor arc of the circle between the points A and B. The tangents to the circle at A and B meet at the point P. If ∠ACB = 102°, then what is the measure of ∠APB?

AB is a chord of a circle in minor segment with center O. C is a point on the minor arc of the circle between the points A and B. The tangents to the circle at A and B meet at the point P. If ∠ACB = 102°, then what is the measure of ∠APB? Correct Answer 24°

Given:

∠ACB = 102°

Concept Used:

When a quadrilateral is inscribed in a circle, the opposite angles of it are supplementary angles.

The angle subtended at the center is always double the angle subtended at the remaining arc.

The radius from the center of the circle to the point of tangency is perpendicular to the tangent line.

Calculation:

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Point D is taken on the major arc of the circle. Then, A & D, B & D, C & B, and A & C are joined.

∠ACB = 102°

Now,

∠ADB = (180 – 102)° = 78°

Now,

∠AOB = (78 × 2)° = 156°

Since OA = OB, ΔAOB is an isosceles triangle and ∠OBA = ∠OAB.

Now,

∠AOB + ∠OBA + ∠OAB = 180°

⇒ 2 × ∠OBA = (180 – 156)°

⇒ ∠OBA = 12°

Now,

∠OBA = 12°

∠OAB = 12°

Now,

∠OAP = ∠OAB + ∠BAP

⇒ 90° = 12° + ∠BAP

⇒ ∠BAP = 78°

Now,

∠OBP = ∠OBA + ∠ABP

⇒ 90° = 12° + ∠ABP

⇒ ∠ABP = 78°

Now,

∠APB + ∠ABP + ∠BAP = 180°

⇒ ∠APB + 78° + 78° = 180°

⇒ ∠APB = 180° - (2 × 78)°

⇒ ∠APB = 24°

∴ The required measure of ∠APB is 24°.

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