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Q.

In an equilateral triangle, the inradius, circumradius and one of the ex-radii are in the ratio

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a

3:7:9

b

2:3:5

c

1 :3:7

d

1:2:3

answer is B.

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Detailed Solution

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we have $Δ = \frac{\sqrt{3}}{4} a^{2}, s = \frac{3 a}{2}$
$\begin{aligned} ∴ r = \frac{Δ}{s} = \frac{a}{2 \sqrt{3}}, R = \frac{a b c}{4 Δ} = \frac{a^{3}}{\sqrt{3} a^{2}} = \frac{a}{\sqrt{3}} \\ and r_{1} = \frac{Δ}{s - a} = \frac{\sqrt{3} / 4 a^{2}}{a / 2} = \frac{\sqrt{3}}{2} a \end{aligned}$
$Hence, r : R : r_{1} = \frac{a}{2 \sqrt{3}} : \frac{a}{\sqrt{3}} : \frac{\sqrt{3}}{2} a = 1 : 2 : 3$

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