Question

In a triangle, if the ex-radii $r_{1}, r_{2}, r_{3}$ are in the ratio $1: 2: 3$, then its sides are in the ratio

• A. 5 : 8 : 9
• B. 5 : 4 : 3
• C. 7 : 9 : 11
• D. 1 : 2 : 3

Answer 1

Answer: (A)

Given that, $r_{1}, r_{2}, r_{3}$ are ex-radii of triangle and $r_{1}: r_{2}: r_{3}=1: 2: 3$
$r_{1}=x, r_{2}=2 x, r_{3}=3 x$
$s-a=\frac{\Delta}{r_{1}} $
$\Rightarrow \,s-a=\frac{\Delta}{x}\,\,\,\,\,\dots(i)$
$s-b=\frac{\Delta}{r_{2}}$
$ \Rightarrow \,s-b=\frac{\Delta}{2 x}\,\,\,\,\,\dots(ii)$
$s-c=\frac{\Delta}{r_{3}}$
$ \Rightarrow \, s-c=\frac{\Delta}{3 x}\,\,\,\,\,\,\dots(iii)$
On adding Eqs. (i), (ii) and (iii), we get
$3 s-(a+b+c)=\frac{\Delta}{x}+\frac{\Delta}{2 x}+\frac{\Delta}{3 x}$
$s=\frac{11 \Delta}{6 x}$
From Eqs. (i), (ii), (iii), we get
$a=\frac{5 \Delta}{6 x^{\prime}} b=\frac{8 \Delta}{6 x^{\prime}}, c=\frac{9 \Delta}{6 x}$
So,$ a: b: c=5: 8: 9 $