Using ruler and compasses only, construct a △ABC such that BC = 5 cm, AB = 6.5 cm and ∠ABC = 120°.

(i) Construct a circumcircle of △ABC.

(ii) Construct a cyclic quadrilateral ABCD such that D is equidistant from AB and BC.

(i) Steps of construction :

1. Draw a line segment AB = 6.5 cm.

2. From B construct angle 120° and extend the line such that BC = 5 cm.

3. Join points A, B and C. Hence, the △ABC is formed.

4. Draw the perpendicular bisector of AB and BC. Let these bisectors meet at the point O.

5. With O as center and radius equal to OA, draw a circle. The circle so drawn passes through the points A, B and C, and is required circumcircle of △ABC.

(ii) We know that locus of point equidistant from two sides is the angle bisector of the angle between the lines.

Steps of construction :

1. Draw the angle bisector of ∠ABC.

2. Mark the point as D where the angle bisector of ∠ABC meets the circumcircle.

3. Join AD and CD.

4. ABCD is the cyclic quadrilateral.