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I need to find the inradius of a triangle with side lengths of $20$, $26$, and $24$.

I know the semiperimeter is $35$, but how do I find the area without knowing the height? Thank you.

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By Heron's Formula the area of a triangle with sidelengths $a,b,c$ is $K = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{1}{2}(a+b+c)$ is the semi-perimeter. You can then use the formula $K = rs$ to find the inradius $r$ of the triangle.

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Solution:

Semiperimeter is given

$$K = RS$$ so

$$ \text{K} = \sqrt{35(15)(9)(11)} = \sqrt{51975}$$

concluding that

$$ R = \frac{\sqrt{51975}}{35} = \frac {3 \sqrt {231}} 7 .$$

Please correct me if I got something wrong.

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