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Tension in a String

Δ ABC is a ri...

Question

$Δ A B C$ is a right angled triangle right angled at $B$ and $B D$ is the altitude from $B$ on side $A C$ . If $B D = 9$ cm and inradii of $Δ A B C, Δ A B D$ and $Δ B C D$ are $r_{1}, r_{2}$ and $r_{3}$ respectively, then $r_{1} + r_{2} + r_{3}$ is

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Solution

In radius
$r_{1} = (s - b) tan \frac{B}{2} = (s - b)$
$= \frac{a - b + c}{2}$
$r_{2} = \frac{x + p - c}{2}, r_{3} = \frac{p + y - a}{2}$
$\Rightarrow r_{1} + r_{2} + r_{3} = \frac{x + y + 2 p - b}{2}$
$∴ r_{1} + r_{2} + r_{3} = p = 9$

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