Question

In the adjoining figure, ‘O’ is the centre of the circle and ΔABC is an equilateral triangle. Find: 

1. ∠AEC 
2. ∠ADС

Answer 1

Given:

O is the centre of the circle.

Triangle ABC is equilateral (each angle = 60°).

Points D and E lie on the circle as shown.

(i) To find ∠AEC:

Since ABC is equilateral, arc BC = 60°.

O is the centre, so arc BC subtends angle BOC at the centre.

Angle BOC = 60° (equal to arc BC).

Considering triangle BOC (isosceles as OB = OC), angle OBC = angle OCB = `(180°−  60°)/2` = 60°.

Triangle BOC is equilateral.

Point E lies on the circle; thus, ∠BEC subtends arc BC.

Angle at circumference ∠BEC = half of arc BC = 30°.

Considering quadrilateral AEBC cyclic, angles ∠AEC and ∠ABC are related.

Since ∠ABC = 60°, ∠AEC = 60°.

(ii) To find ∠ADC:

D lies on the circle.

Since ABC is equilateral, arcs subtended by sides are 60° each.

Arc ADC is 240° (full circle 360° - arc ABC 120°).

The angle at the circumference subtending arc ADC is half of 240°, which is 120°.

Therefore, ∠ADC = 120°.