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Ex Radius

In ABC, R,r,r...

Question

In $△ A B C, R, r, r_{1}, r_{2}, r_{3}$ denote the circumradius, inradius, the exradii opposite to the vertices $A, B, C$ respectively. Given that $r_{1} : r_{2} : r_{3} = 1 : 2 : 3$ .
The sides of the triangle are in the ratio

1 : 2 : 3

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3 : 5 : 7

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1 : 5 : 9

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5 : 8 : 9

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Solution

The correct option is D $5 : 8 : 9$
$\frac{△}{s - a} : \frac{△}{s - b} : \frac{△}{s - c} = 1 : 2 : 3$
Let $\frac{△}{s - a} = \frac{2 △}{s - b} = \frac{3 △}{s - c} = \frac{△}{6 k} \Rightarrow \frac{1}{s - a} = \frac{1}{6 k}, \frac{1}{s - b} = \frac{1}{3 k}, \frac{△}{s - c} = \frac{1}{2 k} \Rightarrow s - a = 6 k, s - b = 3 k, s - c = 2 k \Rightarrow s = 11 k ∴ a = 5 k, b = 8 k, c = 9 k$
Hence, ratio of sides is $5 : 8 : 9$

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