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Question

ABC is an equilateral triangle with side 4 units and a circle inscribed in it. A line segment goes from a vertex C to the midpoint of AB, M. What is the ratio of the length outside the circle to the length inside it of this line segment CM?


  1. 2:5

  2. 1:4

  3. 3:7

  4. 1:3


Solution

The correct option is A

2:5


The figure given by the question is as below

Midpoint of each side is the point where the side 'touches' the circle, i.e., midpoint of CB, i.e., T is the

point where CB touches the circle.

We are asked to find the ratio CP:PM

By power of a point theorem,

CT×CT=CP×CM

i.e.,CT2=CP×CM

i.e.,4=CP×CM                     ........(1)

Also,

CMB is a right angled triangle with MB=2, CB=4

CM=22+42=20     ........(2)

Using (1) & (2)

4=CP20

CP=420

Also,

CM=CP+PM

i.e.,20=420+PM

PM=1620

 Ratio CP:PM=420:1620

                                          =1:4

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