Question

In \(\triangle ABC\), if \(r_1 = 2r_2 = 3r_3\), then find the ratio \(a : b\).

• A. \(3 : 5\)
• B. \(5 : 3\)
• C. \(4 : 5\)
• D. \(5 : 4\)

Hint:

Use formulas for exradii and their relationships to find side ratios.

Answer 1

The Correct Option is D

Step 1: Understand notation
Let \(r_1, r_2, r_3\) be the exradii corresponding to sides \(a, b, c\) respectively. Step 2: Use exradii formula
Exradius \(r_a = \frac{\Delta}{s - a}\), where \(s\) is semi-perimeter and \(\Delta\) area of triangle. Step 3: Given
\[ r_1 = 2 r_2 = 3 r_3 \] Step 4: Form ratios
\[ r_1 : r_2 : r_3 = 1 : \frac{1}{2} : \frac{1}{3} \] Step 5: Use relations to find \(a : b\)
Using relations between sides and exradii, ratio \(a : b = 5 : 4\).