Question

In a $△ABC$, $a:b:c=4:5:6$. 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

The correct option is C

$\frac{16}{7}$

Explanation for the correct option:

Step 1: Find the semi-perimeter and area of triangle

It is given that in a $△ABC$, $a:b:c=4:5:6$.

Let us assume that the common ratio is $k$.

Thus, $a=4k$, $b=5k$ and $c=6k$.

If $s$ is the semi-perimeter of the triangle, then $s=\frac{a+b+c}{2}$.

$$\begin{gathered} \Rightarrow s=\frac{4k+5k+6k}{2} \\ \Rightarrow s=\frac{15k}{2} \end{gathered}$$

Therefore, the area of the triangle is $A=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$.

$$\begin{gathered} \Rightarrow A=\sqrt{\frac{15k}{2}\left(\frac{15k}{2}-4k\right)\left(\frac{15k}{2}-5k\right)\left(\frac{15k}{2}-6k\right)} \\ \Rightarrow A=\sqrt{\frac{15k}{2}\times \frac{7k}{2}\times \frac{5k}{2}\times \frac{3k}{2}} \\ \Rightarrow A=\sqrt{\frac{1575k^4}{16}} \\ \Rightarrow A^2=\frac{1575k^4}{16} \end{gathered}$$

Step 2: Find the ratio of the radii of circumcircle and incircle

The radius of the circumcircle can be given by, $R=\frac{abc}{4A}$.

The radius of the incircle can be given by, $r=\frac{A}{s}$.

Thus, the ratio, $\frac{R}{r}=\frac{{\displaystyle \frac{abc}{4A}}}{{\displaystyle \frac{A}{s}}}$.

$$\begin{gathered} \Rightarrow \frac{R}{r}=\frac{abcs}{4A^2} \\ \Rightarrow \frac{R}{r}=\frac{4k\times 5k\times 6k\times {\displaystyle \frac{15k}{2}}}{4\times {\displaystyle \frac{1575k^4}{16}}} \\ \Rightarrow \frac{R}{r}=\frac{900k^4}{{\displaystyle \frac{1575k^4}{4}}} \\ \Rightarrow \frac{R}{r}=\frac{3600k^4}{{\displaystyle 1575k^4}} \\ \Rightarrow \frac{R}{r}=\frac{16}{{\displaystyle 7}} \end{gathered}$$

Therefore, the ratio of the radius of the circumcircle to that of the incircle is $\frac{16}{7}$.

Hence, the correct option is (C).