Question

The sides of a triangle are in the ratio $3:4:5$, the relation between $r$ and $R$ for the triangle is

• A. $r=\frac{R}{2}$
• B. $r=\frac{2R}{5}$
• C. $r=\frac{R}{5}$
• D. None of these

Answer 1

The correct option is B $r=\frac{2R}{5}$

Let $a=3x,b=4x,c=5x$

learly $a^2+b^2=c^2\Rightarrow \angle C={90}^o$

$\Rightarrow \Delta =\frac{1}{2}ab=6x^2$

And $s=\frac{a+b+c}{2}=6x$

$\therefore r=\frac{\Delta }{s}=x$

Now using sine rule, $\frac{c}{\sin C}=2R\Rightarrow c=2R\Rightarrow R=\frac{5}{2}x$

Hence $r=\frac{2R}{5}$