Question

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

• A. $\frac{15}{4}$
• B. $\frac{11}{5}$
• C. $\frac{16}{7}$
• D. $\frac{16}{3}$

Answer 1

Answer: (C)

Let sides of the triangle are $4x,5x,6x$.
$s=\frac{4x+5x+6x}{2}=\frac{15}{2}x$
$\Delta =\sqrt{\frac{15}{2}x\left(\frac{15}{2}x-4x\right)\left(\frac{15}{2}x-5x\right)\left(\frac{15}{2}x-6x\right)}$
$=\sqrt{\frac{15}{2}x\times \frac{7}{2}x\times \frac{5}{2}x\times \frac{3}{2}x}$
$=\frac{15\sqrt{7}{x}^2}{4}$
Circumradius, $R=\frac{4x\times 5x\times 6x}{4\times \frac{15\sqrt{7}{x}^2}{4}}$
$=\frac{8}{\sqrt{7}}x$
Inradius, $r=\frac{\frac{15\sqrt{7}}{4}{x}^2}{\frac{15}{2}x}$
$=\frac{\sqrt{7}}{2}x$
$\frac{R}{r}=\frac{\frac{8x}{\sqrt{7}}}{\frac{\sqrt{7}x}{2}}=\frac{16}{7}$