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. $15/4$
• B. $11/5$
• C. $16/7$
• D. $16/3$

Answer 1

The correct option is C $16/7$

Let $\frac{a}{4}=\frac{b}{5}=\frac{c}{6}=k$

$R=\frac{abc}{4\Delta }$ and $r=\frac{\Delta }{s}$

$\Rightarrow \frac{R}{r}=\frac{abc}{4\Delta }\times \frac{s}{\Delta }=\frac{sabc}{4{\Delta }^2}$

$=\frac{sabc}{4s(s-a)(s-b)(s-c)}$

$=\frac{4k\times 5k\times 6k}{4(\frac{15}{2}k-4k)(\frac{15}{2}k-5k)(\frac{15}{2}k-6k)}$

$=\frac{5\times 6\times 8}{7\times 5\times 3}=\frac{16}{7}$