In a triangle ABC,if r_1 = 2r_2 = 3r_3,then a | vedclass

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(D) Given,$r_1 = 2r_2 = 3r_3$. Using the formula $r_1 = \frac{\Delta}{s-a}$,$r_2 = \frac{\Delta}{s-b}$,and $r_3 = \frac{\Delta}{s-c}$,we have: $\frac{

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$5 : 4$

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(D) Given,$r_1 = 2r_2 = 3r_3$. Using the formula $r_1 = \frac{\Delta}{s-a}$,$r_2 = \frac{\Delta}{s-b}$,and $r_3 = \frac{\Delta}{s-c}$,we have: $\frac{\Delta}{s-a} = \frac{2\Delta}{s-b} = \frac{3\Delta}{s-c}$. Dividing by $\Delta$,we get $\frac{1}{s-a} = \frac{2}{s-b} = \frac{3}{s-c} = k$ (let). Then $s-a = \frac{1}{k}$,$s-b = \frac{2}{k}$,and $s-c = \frac{3}{k}$. Adding these,$(s-a) + (s-b) + (s-c) = 3s - (a+b+c) = 3s - 2s = s$. So,$s = \frac{1+2+3}{k} = \frac{6}{k}$,which implies $k = \frac{6}{s}$. Now,$s-a = \frac{1}{6/s} = \frac{s}{6}$ $\Rightarrow 6s - 6a = s$ $\Rightarrow 5s = 6a$ $\Rightarrow a = \frac{5s}{6}$. And $s-b = \frac{2}{6/s} = \frac{2s}{6} = \frac{s}{3}$ $\Rightarrow 3s - 3b = s$ $\Rightarrow 2s = 3b$ $\Rightarrow b = \frac{2s}{3} = \frac{4s}{6}$. Therefore,$a : b = \frac{5s}{6} : \frac{4s}{6} = 5 : 4$.

In a $\triangle ABC$,if $r_1 = 2r_2 = 3r_3$,then $a : b =$

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