Properties of Triangles 3 Question 12
12. In a $\triangle A B C, a: b: c=4: 5: 6$. The ratio of radius of the circumcircle to that of the incircle is… .
(1996, 1M)
Show Answer
Answer:
Correct Answer: 12. $\frac{16}{7}$
Solution:
- We have, $R=\frac{a b c}{4 \Delta}$ and $r=\frac{\Delta}{s}$
$$ \begin{aligned} \frac{R}{r} & =\frac{a b c}{4 \Delta} \cdot \frac{s}{\Delta}=\frac{a b c \cdot s}{4 \Delta^{2}} \\ & =\frac{a b c}{4(s-a)(s-b)(s-c)} \end{aligned} $$
But $a: b: c=4: 5: 6$
[given]
$$ \begin{aligned} & \Rightarrow \quad \frac{a}{4}=\frac{b}{5}=\frac{c}{6}=k \\ & \Rightarrow \\ & \text { Now, } \quad s=\frac{1}{2}(a+b+c)=\frac{1}{2}(4 k+5 k+6 x)=\frac{15 k}{2} \\ & \therefore \quad \frac{R}{r}=\frac{(4 k)(5 k)(6 k)}{4 \frac{15 k}{2}-4 k \quad \frac{15 k}{2}-5 k \frac{15 k}{2}-6 k} \\ & =\frac{30 k^{3}}{k^{3} \frac{15-8}{2}} \frac{\frac{15-10}{2}}{\frac{15-12}{2}}=\frac{30 \cdot 8}{7 \cdot 5 \cdot 3}=\frac{16}{7} \end{aligned} $$
