Question

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.

Answer 1

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°