In \(\triangle ABC\), if \(r_1 = 2r_2 = 3r_3\), then find the ratio \(a : b\).
Show Hint
- \(3 : 5\)
- \(5 : 3\)
- \(4 : 5\)
- \(5 : 4\)
The Correct Option is D
Solution and Explanation
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 \(a : b = 5 : 4\).
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