Question

In $△ABC$ if a:b:c = 4 : 5 : 6 find the ratio of radius of circum circle to that of the circle.

Answer 1

$a:b:c=4:5:6$

$\Rightarrow a=4k$

$b=5k$

$c=6k$

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

$\Rightarrow (s-a)=\frac{7k}{2},(s-b)=\frac{5k}{2},(s-c)=\frac{3k}{2}$

$\Rightarrow {△}^2=s(s-a).(s-b).(s-c)$

$=\frac{15k}{2}\times \frac{7k}{2}\times \frac{5k}{2}\times \frac{3k}{2}$

$\Rightarrow △=15\sqrt{7}{\left(\frac{k}{2}\right)}^2$

$\Rightarrow r=\frac{△}{s}=\frac{15\sqrt{7}{\left(\frac{k}{2}\right)}^2}{\frac{15k}{2}}=\sqrt{7}.\frac{k}{2}$

$\Rightarrow R=\frac{abc}{4△}=\frac{4\times 5\times 6\times k^3}{4\times 15\sqrt{7}{\left(\frac{k}{2}\right)}^2}=\frac{8}{\sqrt{7}}.k$

$\Rightarrow \frac{R}{r}=\frac{\frac{8}{\sqrt{7}}.k}{\frac{\sqrt{7}.k}{2}}=\frac{16}{7}$

Hence, the answer is $16:7.$