Question 9
Two chords AB and AC of a circle subtends angles equal to 90∘ and 150∘ respectively at the centre. Find ∠BAC, if AB and AC lie on the opposite sides of the centre.
In ΔBOA
OB = OA [both are the radius of circle]
∠OAB=∠OBA . . . . (i) [angles opposite to equal sides are equal]
In ΔOAB, ∠OBA+∠OAB+∠AOB=180∘ [by angle sum property of a triangle]
⇒ ∠OAB+∠OAB+90∘=180∘ [from Eq (i)]
⇒ 2∠OAB=180∘−90∘
⇒ 2∠OAB=90∘
⇒ ∠OAB=45∘
Now, in ΔAOC AO = OC [both are the radius of a circle]
∴ ∠OCA=∠OAC ............(ii) [angles opposite to equal sides are equal]
Also, ∠AOC+∠OAC+∠OCA=180∘ [by angle sum property of a triangle]
⇒ 150∘+2∠OAC=180∘ [from Eq. (ii)]
⇒ 2∠OAC=180∘−150∘
⇒ 2∠OAC=30∘
⇒ ∠OAC=15∘
∴ ∠BAC=∠OAB+∠OAC=45∘+15∘=60∘