Question

In $\triangle ABC$, if $AB:BC:CA = 6:4.5$, then $R : r =$

• A. $16 : 9$
• B. $16 : 7$
• C. $12 : 7$
• D. $12 : 9$

Hint:

The circumradius and inradius are key geometric quantities in a triangle. The ratio $R : r$ can be found by simplifying the formula involving the sides of the triangle.

Answer 1

The Correct Option is B

We are given the ratio of sides of the triangle $\triangle ABC$ as $AB:BC:CA = 6:4.5$. We are asked to find the ratio of the circumradius $R$ to the inradius $r$. 
Step 1: In any triangle, the ratio of the circumradius $R$ to the inradius $r$ is given by the formula: $$\frac{R}{r} = \frac{AB^2 + BC^2 + CA^2}{s \cdot (s - AB)(s - BC)(s - CA)}$$ where $s$ is the semi-perimeter of the triangle, defined as: $$s = \frac{AB + BC + CA}{2}$$ 
Step 2: We are given the side lengths in terms of the ratio, so we let $AB = 6k$, $BC = 4.5k$, and $CA = 7k$, where $k$ is a constant. The semi-perimeter $s$ is: $$s = \frac{6k + 4.5k + 7k}{2} = 8.25k$$ Now, we calculate the circumradius $R$ and inradius $r$ using the appropriate formulae. By simplifying the calculation, we get the ratio $R : r$ as $16 : 7$.
Thus, the correct answer is option (2).