Question

In △ABC, if a : b : c = 4 : 5 : 6, then the ratio of the circumference to its in radius is

• A. 16:7
• B. 25:11
• C. 5:4
• D. 9:5

Answer 1

The Correct Option is A

We are given a triangle with sides in the ratio 4:5:6. Let's set $a=4k$, $b=5k$, and $c=6k$ for some $k>0$.

1. Calculate the Semi-perimeter:
$s = \frac{a+b+c}{2} = \frac{4k+5k+6k}{2} = \frac{15k}{2}$

2. Compute the Area using Heron's Formula:
$A = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{\frac{15k}{2} \left( \frac{7k}{2} \right) \left( \frac{5k}{2} \right) \left( \frac{3k}{2} \right)}$
$= \frac{k^2}{4}\sqrt{15 \cdot 7 \cdot 5 \cdot 3} = \frac{15k^2\sqrt{7}}{4}$

3. Find the Inradius:
Using $A = rs$, we get:
$r = \frac{A}{s} = \frac{\frac{15k^2\sqrt{7}}{4}}{\frac{15k}{2}} = \frac{k\sqrt{7}}{2}$

4. Calculate the Circumradius:
Using $R = \frac{abc}{4A}$:
$R = \frac{(4k)(5k)(6k)}{4 \cdot \frac{15k^2\sqrt{7}}{4}} = \frac{120k^3}{15k^2\sqrt{7}} = \frac{8k}{\sqrt{7}}$

5. Compute the Ratio of Circumradius to Inradius:
$\frac{R}{r} = \frac{\frac{8k}{\sqrt{7}}}{\frac{k\sqrt{7}}{2}} = \frac{16}{7}$

Final Answer:
The ratio of the circumradius to the inradius is ${\dfrac{16}{7}}$