Line XY touches the circle with centre O at point A, with X and Y on opposite sides of A. Seg AB is a chord of the circle and ∠BAY = 65°. Point C lies on the major arc AB. Seg OA, seg OB, seg CA and seg CB are drawn.[Figure: a circle with centre O; t…

Q.5 (2) Line XY touches the circle with centre O at point A, with X and Y on opposite sides of A. Seg AB is a chord of the circle and ∠BAY = 65°. Point C lies on the major arc AB. Seg OA, seg OB, seg CA and seg CB are drawn.

[Figure: a circle with centre O; the tangent XY touches it at A at the bottom; chord AB goes up to the right, so the minor arc AB lies inside ∠BAY on the Y side; point C is on the far (major) arc; radii OA and OB are drawn.]

(a) Find m(minor arc AB) and m(major arc AB).
(b) Find m∠ACB.
(c) Find m∠AOB and m∠OAB.

Doubtly AI Asked 27 Oct 2026 AI-generated · not yet verified ssc 10th Save WhatsApp Report

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Answer: (a) 130°, 230° (b) 65° (c) ∠AOB = 130°, ∠OAB = 25°

(a) ∠BAY is formed by the tangent AY and the chord AB, and it intercepts the minor arc AB.

By the tangent-secant angle theorem, m∠BAY = ½ m(minor arc AB), so m(minor arc AB) = 2 × 65° = 130°.

m(major arc AB) = 360° − 130° = 230°.

(b) ∠ACB is an inscribed angle intercepting the minor arc AB, so m∠ACB = ½ × 130° = 65°.

Radius OA ⊥ tangent XY (tangent theorem), so ∠OAY = 90°. Hence ∠OAB = ∠OAY − ∠BAY = 90° − 65° = 25°.

Check: Δ OAB is isosceles (OA = OB), so ∠OAB = ∠OBA = (180° − 130°) ÷ 2 = 25°.

Doubtly AI · 27 Oct 2026 AI-generated · not yet verified Report

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Line XY touches the circle with centre O at point A, with X and Y on opposite sides of A. Seg AB is a chord of the circle and ∠BAY = 65°. Point C lies on the major arc AB. Seg OA, seg OB, seg CA and seg CB are drawn.[Figure: a circle with centre O; t…

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