Question

If $A$ is the area of a right triangle and $b$ be one of the sides containing the right angle, prove that lengths of the altitude on the hypotenuse is $\frac{2Ab}{\sqrt{b^4+4A^2}}.$

Answer 1

R.E.F. Image.

Area $A=\frac{1}{2}\left(AB\right)\left(BC\right)$

$\Rightarrow AB=$ $\left(\frac{2A}{b}\right)$

By Pythagoras theorem,

$A{C}^2=A{B}^2+B{C}^2$

$A{C}^2=b^2+\frac{4A^2}{b^2}$

$AC=\frac{1}{b}\sqrt{b^4+4A^2}$

Now area, A = $\frac{1}{2}\left(BD\right)\left(AC\right)$

$\Rightarrow A=\frac{1}{2b}\sqrt{b^4+4A^2}\left(BD\right)$

$\Rightarrow BD=\frac{2Ab}{\sqrt{b^4+4A^2}}$ Proved