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Question

# â–³ABC 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

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B

2:5

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C

1:4

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D

3:7

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Solution

## 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×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=CP√20 CP=4√20 Also, CM=CP+PM i.e.,√20=4√20+PM ⇒PM=16√20 ∴ Ratio CP:PM=4√20:16√20 =1:4

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