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?

• A. 1:3
• B. 2:5
• C. 1:4
• D. 3:7

Answer 1

The correct option is B

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,

$CT\times CT=CP\times CM$

$i.e.,C{T}^2=CP\times CM$

$i.e.,4=CP\times CM$ ........(1)

Also,

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

$\therefore CM=\sqrt{2^2+4^2}=\sqrt{20}$ ........(2)

Using (1) $\&$ (2)

$4=CP\sqrt{20}$

$CP=\frac{4}{\sqrt{20}}$

Also,

CM=CP+PM

$i.e.,\sqrt{20}=\frac{4}{\sqrt{20}}+PM$

$\Rightarrow PM=\frac{16}{\sqrt{20}}$

$\therefore RatioCP:PM=\frac{4}{\sqrt{20}}:\frac{16}{\sqrt{20}}$

$=1:4$