Question

In $\Delta$ $ABC$ the sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively.

If the sides of a triangle are in the ratio $3:7:8$,

then $R:r$ is equal to?

• A. $2:7$
• B. $7:2$
• C. $3:7$
• D. $7:3$

Answer 1

The correct option is B $7:2$

Given, $a=3k,b=7k,c=8k$

$r=4R\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$

$\Rightarrow \frac{R}{r}=\frac{1}{4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}}$

$=\frac{1}{4\sqrt{\frac{(s-b)(s-c)}{bc}\times \frac{(s-a)(s-c)}{ac}\times \frac{(s-a)(s-b)}{ab}}}$

$=\frac{abc}{4(s-a)(s-b)(s-c)}=\frac{\left(3k\right)\left(7k\right)\left(8k\right)}{4\left(6k\right)\left(2k\right)\left(k\right)}=\frac{7}{2}\therefore R:r=7:2$