Question
In a circle of radius 13 cm, a chord has length 10 cm.
Find: (a) the distance of the chord from the centre, (b) the angle subtended by the chord at the centre (in degrees, correct to nearest degree).Solution
(a) Let distance from centre O to chord AB be d, and M be midpoint of AB. OM ⟂ AB, AM = 10/2 = 5 cm Right triangle OMA: OA² = OM² + AM² 13² = d² + 5² 169 = d² + 25 ⇒ d² = 144 ⇒ d = 12 cm (b) Let central angle ∠AOB = θ. Chord length: AB = 2R sin(θ/2) 10 = 2×13×sin(θ/2) 10 = 26 sin(θ/2) sin(θ/2) = 10/26 = 5/13 θ/2 = sin⁻¹(5/13) ≈ 22.62° ⇒ θ ≈ 45.24° So angle ≈ 45° (nearest degree).
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