Two chords AB and AC of a circle subtends angles equal to 90° and 150°, respectively at the center. Find ∠BAC, if AB and AC lie on the opposite sides of the center.

Question

Two chords AB and AC of a circle subtends angles equal to 90° and 150°, respectively at the center. Find ∠ BAC, if AB and AC lie on the opposite sides of the center.

Philoid · Accepted Answer

In ΔAOB, OA = OB (both are radius of the circle)

∴ ∠OBA = ∠OAB (angle opposite to equal sides are equal)

Using the angle sum property in ΔAOB, sum of all angles of the triangle is 180°, we have:

∠OAB + ∠AOB +∠OBA = 180°

⇒ ∠OAB +90° + ∠OAB = 180°

⇒ 2∠OAB = 180° - 90°

⇒ 2∠OAB = 90°

⇒ ∠OAB = 45°

Now, in ΔAOC,

OA = OC (both are radius of the circle)

∴ ∠OCA = ∠OAC (angle opposite to equal sides are equal)

Using the angle sum property in ΔAOB, sum of all angles of the triangle is 180°, we have:

∠OAC + ∠AOC +∠OCA = 180°

⇒ ∠OAC +150° + ∠OAC = 180°

⇒ 2∠OAC = 180° - 150°

⇒ 2∠OAC = 30°

⇒ ∠OAC = 15°

Now, ∠BAC = ∠OAB + ∠OAC

= 45° + 15°

= 60°

∴ ∠BAC = 60°