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प्रश्न
Two chords AB and AC of a circle subtends angles equal to 90º and 150º, respectively at the centre. Find ∠BAC, if AB and AC lie on the opposite sides of the centre.
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उत्तर
In triangle BOA,
OB = OA ...[Both are the radius of circle]
∠OAB = ∠OBA ...(i) [Angle opposite to equal sides are equal]
Now, In triangle OAB,
∠OBA + ∠AOB + ∠AOC = 180° ...[By angle sum property of a triangle]
∠OAB + ∠OAB + 90° = 180° ...[From equation (i)]
2∠OAB = 180° – 90°
2∠OAB = 90°
∠OAB = 45°
Again, in triangle AOC,
AO = OC ...[Radii or circle]
∠OCA = ∠OAC ...(ii) [Angle opposite to equal sides are equal]
Now, by angle sum property of a triangle,
∠AOC + ∠OAC + ∠OCA = 180°
150° + 2∠OAC = 180° ...[From equation (ii)]
2∠OAC = 180° – 150°
2∠OAC = 30°
∠OAC = 15°
∠BAC = ∠OAB + ∠OAC = 45° + 15° = 60°
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