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# Find the shortest altitude

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In triangle $ABC,$ $AB = 10,$ $BC = 24,$ and $AC = 26.$ Find the length of the shortest altitude in this triangle.

Mar 29, 2020

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In triangle $ABC,$ $AB = c = 10,$ $BC = a = 24,$ and $AC = b = 26.$ Find the length of the shortest altitude in this triangle.

Hello Guest!

$$h_a:h_b:h_c=\frac{1}{a}:\frac{1}{b}:\frac{1}{c}=\frac{1}{24}:\frac{1}{26}:\frac{1}{10}$$

$$h_b$$ is the shortest altitude.

$$h_b=\frac{2}{b}\cdot \sqrt{s(s-a)(s-b)(s-c)}$$

$$s=\frac{1}{2}(a+b+c)=\frac{1}{2}(24+26+10)\\ s=30$$

$$h_b=\frac{2}{26}\cdot \sqrt{30(30-24)(30-26)(30-10)}$$

$$h_b=9.231$$

$$h_b =9.231$$ is the shortest altitude.

!

Mar 29, 2020
edited by asinus  Mar 29, 2020