Question

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

In a $\Delta ABC,$ the sides are in the ratio $4:5:6$. The ratio of the circumradius and the inradius is

• A. $8:7$
• B. $3:2$
• C. $7:3$
• D. $16:7$

Answer 1

The correct option is D $16:7$

Let the sides be $4k,5k,6k$

$s=\frac{15k}{2}$

$\Delta =\sqrt{\frac{15k}{2}\left(\frac{7k}{2}\right)\left(\frac{5k}{2}\right)\left(\frac{3k}{2}\right)}$

$R=\frac{abc}{4\Delta }=\frac{120k^3}{4\Delta }$

$r=\frac{\Delta }{s}=\frac{2\Delta }{15k}$

$\Rightarrow \frac{R}{r}=\frac{\frac{120k^3}{4\Delta }}{\frac{2\Delta }{15k}}=\frac{225k^4}{{\Delta }^2}$

$\Rightarrow \frac{R}{r}=\frac{225k^4}{\frac{225\times 7k^4}{16}}=\frac{16}{7}$