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° with each other, from a point E.
Refer to the figure and select the option which would lead you to the required construction. The distance d is the distance OE.

• A. Using trigonometry, arrive at d = r and mark E.
• B. Construct the $△$MNO as it is equilateral triangle.
• C. Mark M and N on the circle such that $\angle$MOE = ${60}^{\circ }$ and $\angle$NOE = ${60}^{\circ }$.
• D. Mark M and N on the circle such that $\angle$MOE = ${120}^{\circ }$ and $\angle$NOE = ${120}^{\circ }$.

Answer 1

The correct option is C

Mark M and N on the circle such that $\angle$MOE = ${60}^{\circ }$ and $\angle$NOE = ${60}^{\circ }$.

Since the angle between the tangents is 60°, we get $\angle MON={120}^{\circ }$
(As MONE is a quadrilateral and sum of angles of a quadrilateral is ${360}^{\circ }$).
Hence, ΔMNO is NOT equilateral.

Since E is outside the circle, d can not be equal to r.

We know that ∠MOE = 60°, following are the steps of construction:

1. Draw a ray from the centre O.

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

3. Now extend OM and from M, draw a line perpendicular to OM. This intersects the ray at E. This is the point from where the tangents should be drawn and EM is one tangent.

4. Similarly, EN is another tangent.