Question

In a triangle $a:b:c=4:5:6$. Find the ratio of the radius of the circumcircle to that of the incircle.

• A. $\frac{8}{7}$
• B. $\frac{16}{7}$
• C. $\frac{4}{7}$
• D. none of these

Answer 1

The correct option is B $\frac{16}{7}$

Let $a=4\lambda ,b=5\lambda$ and $c=6\lambda$

Now $\frac{R}{r}=\frac{abc}{4\Delta r}=\frac{sabc}{4{\Delta }^2}[\because r=\frac{\Delta }{s}]$

$=\frac{\left(4\lambda .5\lambda .6\lambda \right)\left(\frac{15}{2}\lambda \right)}{4\left(s\right)(s-a)(s-b)(s-c)}$

$=\frac{900{\lambda }^2}{4\left(\frac{15}{2}\lambda \right)\left(\frac{7}{2}\lambda \right)\left(\frac{5}{2}\lambda \right)\left(\frac{3}{2}\lambda \right)}=\frac{16}{7}$