Question

In a triangle $ABC,a:b:c=4:5:6$. The ratio of radius of circum to the incircle

• A. $7:3$
• B. $16:3$
• C. $16:7$
• D. $noneofthese$

Answer 1

Answer: (C)

$16:7$

Given,

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

$\therefore a=4k,b=5k,c=6k$

perimeter of $△=s=\frac{a+b+c}{2}=\frac{4k+5k+6k}{2}=\frac{15k}{2}$

Ar of $△=\sqrt{s(s-a)(s-b)(s-c)}=\frac{15k}{2}\times \frac{7k}{2}\times \frac{5k}{2}\times \frac{1k}{2}=15\sqrt{7}{\left(\frac{k}{2}\right)}^2$

inradius $r=\frac{Arof△}{perimeter}=\frac{15\sqrt{7}{\left(\frac{k}{2}\right)}^2}{\frac{15k}{2}}=\sqrt{7}\frac{k}{2}$

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

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