Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60°.

Steps of construction:

1. With O as the centre and 5cm as radius draw a circle.
2. Take a point A on the circumference of the circle and join OA.
3. Draw AX perpendicular to OA.
4. Construct ∠AOB = 120° where B lies on the circumference.
5. Draw BY perpendicular to OB.
6. Both AX and BY intersect at P.
7. PA and PB are the required tangents inclined at 60°.

Proof:

∠OAP = ∠OBP = 90° (By construction)

∠AOB = 120° (By construction)

In quadrilateral OAPB,

∠APB = 360° - [∠OAP + ∠OBP + ∠AOB]

= 360° - [90° + 90° + 120°]

= 360° - 300°

= 60°

Hence PA and PB are the required tangents inclined at 60°.