Question

If $\mathrm{A}$ is the area of a right-angled triangle and $\mathrm{B}$ is one of the sides containing the right angle, then calculate the length of altitude on the hypotenuse?

Answer 1

Compute the required length:

Given : The area of a right triangle is $\mathrm{A}$. and One side of the right angle is $\mathrm{B}$.

Let base of the triangle is $B$ and height is $h$.

We know that area of triangle is given by,

$A=\frac{1}{2}\times \left(Base\right)\times \left(Height\right)$

$$\begin{gathered} \Rightarrow A=\frac{1}{2}\times B\times h \\ \Rightarrow h=\frac{2A}{B} \end{gathered}$$

Now, using Pythagoras's theorem

${\left(\mathrm{Hypotenuse}\right)}^2=(\mathrm{Base}{)}^2+{\left(\mathrm{Height}\right)}^2$

$$\begin{gathered} \Rightarrow {\left(\mathrm{Hypotenuse}\right)}^2={\left(B\right)}^2+{\left(\frac{2\mathrm{A}}{B}\right)}^2 \\ \Rightarrow \mathrm{Hypotenuse}=\sqrt{{\left(B\right)}^2+{\left(\frac{2\mathrm{A}}{B}\right)}^2} \\ \Rightarrow \mathrm{Hypotenuse}=\sqrt{\left(\frac{{\left(B\right)}^4+{\left(2\mathrm{A}\right)}^2}{B^2}\right)} \\ \Rightarrow \mathrm{Hypotenuse}=\frac{1}{B}\sqrt{{\left(B\right)}^4+{\left(2\mathrm{A}\right)}^2} \end{gathered}$$

Now, using the area of the right-angled triangle

$$\begin{gathered} \mathrm{A}=\frac{1}{2}\times \frac{1}{B}\sqrt{{\left(B\right)}^4+{\left(2\mathrm{A}\right)}^2}\times H \\ \Rightarrow 2AB=\sqrt{{\left(B\right)}^4+{\left(2\mathrm{A}\right)}^2}\times H \\ \Rightarrow H=\frac{2AB}{\sqrt{{\left(B\right)}^4+{\left(2\mathrm{A}\right)}^2}} \end{gathered}$$

Hence, the length of the altitude to the hypotenuse is $\frac{2AB}{\sqrt{{\left(B\right)}^4+{\left(2\mathrm{A}\right)}^2}}$..