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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. Refer to the figure and select the option which would lead us to the required construction. 'd' is the distance OE.


 


  1. Construct the ΔMNO as it is equilateral

  2. Mark M and N on the circle such that MOE=60 and NOE=60

  3. Using trigonometry, arrive at d=r(3) and mark E.

  4. None of the above


Solution

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 and OE bisects MENMEO=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,sin(30) =12=OMOE=rd we have d = 2r and hence option (C) is also ruled out.

Recalling that MOE=60, following are the steps of construction:
1. Draw a ray OX from the centre. This ray contains the point (E) from which tangents are drawn.
2. With O as centre, construct MOX=60 
3. 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.

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