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).