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 $r_1:r_2:r_3=1:2:3$
$r_1=x,r_2=2x,r_3=3x$
$s-a=\frac{\Delta }{r_1}$
$\Rightarrow s-a=\frac{\Delta }{x}\dots (i)$
$s-b=\frac{\Delta }{r_2}$
$\Rightarrow s-b=\frac{\Delta }{2x}\dots (ii)$
$s-c=\frac{\Delta }{r_3}$
$\Rightarrow s-c=\frac{\Delta }{3x}\dots (iii)$
On adding Eqs. (i), (ii) and (iii), we get
$3s-(a+b+c)=\frac{\Delta }{x}+\frac{\Delta }{2x}+\frac{\Delta }{3x}$
$s=\frac{11\Delta }{6x}$
From Eqs. (i), (ii), (iii), we get
$a=\frac{5\Delta }{6x^{\prime }}b=\frac{8\Delta }{6x^{\prime }},c=\frac{9\Delta }{6x}$
So,$a:b:c=5:8:9$