Question

You are given a circle with radius 'r' and centre O. You are asked to draw a pair of tangents from a point E which are inclined at an angle of 120° to each other. Refer the figure and select the option which would lead us to the required construction.

• A. Using trigonometry, arrive at d = r and mark E.
• B. Construct the $△$ MNO as it is equilateral
• C. Mark M and N on the circle such that $\angle$MOE = ${60}^{\circ }$ and $\angle$NOE = ${60}^{\circ }$
• D. None of these

Answer 1

The correct option is B

Construct the $△$ MNO as it is equilateral

Since the angle between the tangents is 120° and OE bisects ∠MEN, ∠MEO = 60°. ΔOME is a right angled triangle, right angled at M, therefore ∠MOE = 30°. Similarly ∠NOE = 30° and hence ∠MON = 60°.

Also, MO = ON $\Rightarrow$ ∠OMN = ∠ONM. Thus, ∠OMN = ∠ONM = ∠MON = 60°. Hence ΔMNO is equilateral.

Following are the steps of construction:

1. Join O and a point on the circle. Mark the point as M.

2. With O as centre, construct ∠MON = 60° .

3. Join M and N. Construct 90° at M and N.

4. Extend the lines till they meet a point E.

5. EM and EN are the required tangents.