Question

If A be the area of a right triangle and b is one of the sides containing the right angle. Find $m$ if the length of the altitude on the hypotenuse is $\frac{mAb}{\sqrt{b^4+4A^2}}$ .

Answer 1

Area of triangle $=A=\frac{1}{2}\times \left(base\right)\times \left(height\right)$

One side containing Right angle $=b$

Let another side $=x$

So, area $=A=\frac{1}{2}\times b\times x$

$\Rightarrow x=\frac{2A}{b}$

From Pythagoras theorem:

Hypotenuse $=\sqrt[2]{2b^2+x^2}$

$=\sqrt[2]{2b^2+\frac{4A^2}{b^2}}$

$=\frac{\sqrt[2]{2b^4+4A^2}}{b}$

Length of altitude on the hypotenuse $=\frac{m\times A\times b}{\sqrt[2]{2b^4+4A^2}}$

Area of triangle $=A=\frac{1}{2}$x(hypotenuse)x(length of altitude on on hypotenuse)

$A=\frac{\sqrt[2]{2b^4+4A^2}\times m\times A\times b}{\sqrt[2]{2b^4+4A^2}\times 2b}$

$\Rightarrow m=2$