Question

In a triangle ABC, if \(r_1 = 3, r_2 = 4, r_3 = 6\), then \(b =\)

• A. \(2\sqrt{6}\)
• B. \(\dfrac{5\sqrt{6}}{3}\)
• C. \(\dfrac{7\sqrt{6}}{3}\)
• D. \(3\sqrt{6}\)

Hint:

Use the exradius formula \(r = \dfrac{\Delta}{s - \text{side}}\) and manipulate equations accordingly to find unknown sides.

Answer 1

The Correct Option is A

Given exradii \(r_1, r_2, r_3\), use relation with triangle sides: \[ r_1 = \dfrac{\Delta}{s-a}, \quad r_2 = \dfrac{\Delta}{s-b}, \quad r_3 = \dfrac{\Delta}{s-c}. \] By computing and equating appropriate expressions, we find \(b = 2\sqrt{6}\).