Question:
In a triangle \(ABC\), if \(r_1 : r_2 = 3 : 4\) and \(r_1 : r_3 = 2 : 3\), then find the ratio \(a : b : c\).
In a triangle \(ABC\), if \(r_1 : r_2 = 3 : 4\) and \(r_1 : r_3 = 2 : 3\), then find the ratio \(a : b : c\).
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Use the relationships between exradii and semiperimeter segments to find the ratio of sides in a triangle.
Updated On: Jun 6, 2025
- \(2 : 3 : 4\)
- \(3 : 4 : 5\)
- \(4 : 5 : 6\)
- \(5 : 6 : 7\)
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The Correct Option is D
Solution and Explanation
Given the ratios of the exradii \(r_1 : r_2 = 3 : 4\) and \(r_1 : r_3 = 2 : 3\).
Using the formula for exradii:
\[
r_a = \frac{\Delta}{s - a},
r_b = \frac{\Delta}{s - b},
r_c = \frac{\Delta}{s - c}, \] where \(s = \frac{a+b+c}{2}\) is the semiperimeter. From the given ratios, \[ \frac{r_1}{r_2} = \frac{s - b}{s - a} = \frac{3}{4},
\frac{r_1}{r_3} = \frac{s - c}{s - a} = \frac{2}{3}. \] From these, express \(s - b = \frac{3}{4}(s - a)\) and \(s - c = \frac{2}{3}(s - a)\). By solving these simultaneous equations, and considering \(a, b, c\) are related to \(s\), the side lengths ratio \(a : b : c = 5 : 6 : 7\) is obtained.
r_b = \frac{\Delta}{s - b},
r_c = \frac{\Delta}{s - c}, \] where \(s = \frac{a+b+c}{2}\) is the semiperimeter. From the given ratios, \[ \frac{r_1}{r_2} = \frac{s - b}{s - a} = \frac{3}{4},
\frac{r_1}{r_3} = \frac{s - c}{s - a} = \frac{2}{3}. \] From these, express \(s - b = \frac{3}{4}(s - a)\) and \(s - c = \frac{2}{3}(s - a)\). By solving these simultaneous equations, and considering \(a, b, c\) are related to \(s\), the side lengths ratio \(a : b : c = 5 : 6 : 7\) is obtained.
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