Draw a pair of tangents to a circle of radius 4.5 cm, which are inclined to each other at an angle of 60°.

Given angle between tangents is 60°.

i.e. ∠ ADC = 60°

Since angle subtended is double the angle between tangents.

∠AOC = 2× 60° = 120°

So, we need to draw ∠AOC = 120°

∴ we draw a radius, then second radius at 120° from first.

Also,

Tangent is perpendicular to radius.

So,

OA ⊥ AB and OC⊥ AC.

Thus, to make tangents, we draw perpendiculars from A and C.

Steps of construction:

1. Draw a circle of horizontal radius OA = 4.5 cm.

2. Draw angle of 120° from point O which intersects at C.

3. Draw 90° from point A.

4. Draw 90° from C. When two arcs intersect mark it as point D.

Both tangents AD and CD intersect at 60°.

Justification:

We need to prove that AD and CD are the tangents to the circle at 60°.

Since ∠DAO = 90°

∴ DA ⊥ OA

Since tangent is perpendicular to radius and OA is the radius.

∴ DA is the tangent to the circle.

Similarly,

CD is the tangent.

Now we must prove ∠D = 60°.

In quadrilateral AOCD,

∠A + ∠O + ∠C + ∠D = 360°

90° + 120° + 90° + ∠D = 360°

300° + ∠D = 360°

∠D = 60°

Hence, AD and CD are the tangents to the circle at 60°.