Question

You are given a circle with radius ‘r’ and centre O. You are asked to draw a pair of tangents which are inclined at an angle of ${60}^{\circ }$ to each other. Refer the figure and select the option which would lead us to the required construction. d is the distance OE.

• A. Construct the ΔMNO as it is equilateral
• B. Mark M and N on the circle such that ∠MOE = 60° and ∠NOE = 60°
• C. Using trigonometry, arrive at d = r√3 and mark E.
• D. Using trigonometry, arrive at d = r√5 and mark E.

Answer 1

The correct option is B

Mark M and N on the circle such that ∠MOE = 60° and ∠NOE = 60°

Since the angle between the tangents is ${60}^{\circ }$ and OE bisects $\angle$MEN, $\angle$MEO = ${30}^{\circ }$ .

$△$OME is a right angled triangle, right angled at M, therefore $\angle$MOE = ${60}^{\circ }$. Similarly $\angle$NOE =${60}^{\circ }$ and hence $\angle$MON = ${120}^{\circ }$.

Following are the steps of construction:

1. Draw a ray from the centre O.

2. With O as centre, construct $\angle$MOE = ${60}^{\circ }$

3. Now extend OM and at M, draw a line perpendicular to OM. This intersects the ray at E. EM is one of the required tangents.

4. Similarly tangent EN can be drawn.