Question

In a triangle $A B C, a: b: c=4: 5: 6 .$ 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 $4 x, 5 x, 6 x$.
$s =\frac{4 x+5 x+6 x}{2}=\frac{15}{2} x$
$\Delta =\sqrt{\frac{15}{2} x\left(\frac{15}{2} x-4 x\right)\left(\frac{15}{2} x-5 x\right)\left(\frac{15}{2} x-6 x\right)}$
$=\sqrt{\frac{15}{2} x \times \frac{7}{2} x \times \frac{5}{2} x \times \frac{3}{2} x}$
$=\frac{15 \sqrt{7} x^{2}}{4}$
Circumradius, $R =\frac{4 x \times 5 x \times 6 x}{4 \times \frac{15 \sqrt{7} x^{2}}{4}}$
$=\frac{8}{\sqrt{7}} x$
Inradius, $r=\frac{\frac{15 \sqrt{7}}{4} x^{2}}{\frac{15}{2} x}$
$=\frac{\sqrt{7}}{2} x$
$\frac{R}{r} =\frac{\frac{8 x}{\sqrt{7}}}{\frac{\sqrt{7} x}{2}}=\frac{16}{7}$