Question

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

Options

• Using trigonometry, arrive at d = `sqrt5` and mark E.

• Using trigonometry, arrive at d = `sqrt3` and mark E.

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

• Construct the ΔMNO as it is equilateral

Answer 1

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

Explanation:-

Since the angle between the tangents is 60° and OE bisects ∠MEN, ∠MEO = 30°.

Now, since ΔOME is a right angled triangle, right angled at M, we realise that the ∠MOE = 60°. Since ∠MOE = 60°, we must have ∠NOE =60° and hence ∠MON = 120°. Hence ΔMNO is NOT equilateral.

Next, since in ΔOME, sin30°= `1/2 = (OM)/(OE) = r/d,` we have d = 2r.

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

1. Draw a ray from the centre O.
2. With O as centre, construct ∠MOE = 60° [constructing angle 60° is easy]
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, EM is one tangent.
4. Similarly, EN is another tangent.