lf $r_{1} = 8, r_{2} = 12, r_{3} = 24$ , then $C =$

\frac{π}{4}

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\frac{π}{6}

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\frac{π}{3}

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\frac{π}{2}

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Solution

The correct option is C $\frac{π}{2}$
$r_{1} = \frac{Δ}{s - a}, r_{2} = \frac{Δ}{s - b}, r_{3} = \frac{Δ}{s - c}$
$\frac{r_{1}}{r_{2}} = \frac{s - b}{s - a} = \frac{8}{12}, \frac{r_{2}}{r_{3}} = (\frac{s - c}{s - b}) = \frac{12}{24}$
$\frac{r_{3}}{r_{1}} = (\frac{s - a}{s - c}) = \frac{24}{8}$
$\frac{s - b}{s - a} = \frac{2}{3}, \frac{s - c}{s - b} = \frac{1}{2}, \frac{s - a}{s - c} = 3$
$\frac{a + c - b}{b + c - a} = \frac{2}{3}, \frac{a + b - c}{a + c - b} = \frac{1}{2}, \frac{b + c - a}{a + b - c} = 3$
$4 a + 2 b = 4 c$
$2 a + b = 2 c$
$5 a + c = 5 b - (1), a + 3 b = 3 c - (2)$ ...now from these two equations we get,..(substitute the value c from (1) to (2)).
$a + 3 b = 15 b - 15 a$
$16 a = 12 b$
$4 a = 3 b$
$c = 5 b - 5 a$
$c = \frac{5}{4} b$
$∴ a : b : c = 3 : 4 : 5$
$∴ ∠ c = π / 2$

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