Question

Fill In The Blanks

Two chords AB and AC of a circle are on the opposite sides of the centre. If AB and AC subtend angles equal to 90^∘ and 150^∘ respectively at the centre, then ∠BAC = _________.

Answer 1

Given:
AB and AC subtend angles equal to 90° and 150° respectively at the centre
i.e., ∠AOB = 90° and ∠AOC = 150° ...(1)

In ∆AOB,
OA = OB (radius)
∴ ∠OBA = ∠OAB (angles opposite to equal sides are equal) ...(2)

Now, ∠OBA + ∠OAB + ∠AOB = 180° (angle sum property)
⇒ 2∠OAB + 90° = 180° (From (1) and (2))
⇒ 2∠OAB = 180° − 90°
⇒ 2∠OAB = 90°
⇒ ∠OAB = 45° ...(3)


In ∆AOC,
OA = OC (radius)
∴ ∠OCA = ∠OAC (angles opposite to equal sides are equal) ...(4)

Now, ∠OCA + ∠OAC + ∠AOC = 180° (angle sum property)
⇒ 2∠OAC + 150° = 180° (From (1) and (4))
⇒ 2∠OAC = 180° − 150°
⇒ 2∠OAC = 30°
⇒ ∠OAC = 15° ...(5)


Thus,
∠BAC = ∠OAC + ∠OAB
⇒ ∠BAC = 15° + 45° (From (3) and (5))
⇒ ∠BAC = 60°

Hence, ∠BAC = 60°.