Question

In the adjoining diagram, chords AB, BC and CD are equal. O is the centre of the circle. If ∠ ABC = 120°, Calculate: (i) ∠ BAC,   (ii) ∠ BEC, (iii) ∠ BED, (iv) ∠ COD

Answer 1

(i)
In Δ ABC,
∠ ABC + ∠ BAC + ∠ BCA = 180°  ....( ∵ The sum of three angles of a triangle is 180°)
120° + ∠ BAC + ∠ BCA = 180°     ....( ∵ ∠ ABC = 120° (Given))
∠ BAC + ∠ BCA = 60° 

But, BA = BC.
∠ BAC + ∠ BCA = 60° 
2 ∠ BCA = 60° 
∠ BCA = 30°

(ii) ∠ BEC = ∠ BAC = 30°

(iii)
AB = BC = CD
Arc AB = Arc BC = Arc CD
Now,
∠ COB = 2 ∠ CAB
∠ COB = 2 x 30° = 60°
∠ DOC = ∠ COB = 60°
∠ DEC = `1/2 "∠ DOC" = 1/2 xx 60° = 30° `

∴ ∠ BED = ∠ BEC + ∠ DEC
∠ BED = ∠ BAC + ∠ DEC
∠ BED = 30° + 30° = 60°

(iv) ∠ COD = 60°