Question
In a circle of radius 10 cm, a chord AB has length
10√3 cm. Find: (i) the distance of the chord from the centre, and (ii) the measure of ∠AOB, where O is the centre.Solution
(i) Half of chord AB = (10√3)/2 = 5√3 cm. Let OM be perpendicular from centre O to AB, with M the midpoint. Then triangle OMB is right-angled. OB = 10, MB = 5√3, OM = d. OB² = OM² + MB² 10² = d² + (5√3)² 100 = d² + 75 ⇒ d² = 25 ⇒ d = 5 cm Distance of chord from centre = 5 cm. (ii) For central angle θ = ∠AOB, chord length formula: AB = 2r sin(θ/2) 10√3 = 2 × 10 × sin(θ/2) 10√3 = 20 sin(θ/2) sin(θ/2) = (10√3)/20 = √3/2 ⇒ θ/2 = 60° ⇒ θ = 120°
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