Question

If $A$ be the area of a right angled triangle and $b$ is the length of one of the sides containing the right angle, then the length of the altitude on the hypotenuse is

• A. $\frac{2Ab}{\sqrt{b^2+4A^2}}$
• B. $\frac{2Ab}{b^2+4A^2}$
• C. $\frac{2Ab}{\sqrt{b^4+4A^4}}$
• D. $\frac{2Ab}{\sqrt{b^4+4A^2}}$

Answer 1

The correct option is D $\frac{2Ab}{\sqrt{b^4+4A^2}}$

Let the length of the other side and the altitude on the hypotenuse be $x$ and $p$ respectively. Then,
$A=\frac{1}{2}\times b\times x$ and $A=\frac{1}{2}\times p\times AB$

$\Rightarrow x=\frac{2A}{b}$ and $p=\frac{2A}{AB}$
$A{B}^2=x^2+b^2$
$\Rightarrow AB=\sqrt{x^2+b^2}=\sqrt{{\left(\frac{2A}{b}\right)}^2+b^2}$
$\therefore p=\frac{2A}{\sqrt{\left(\frac{2A}{b}\right)+b^2}}=\frac{2A}{\sqrt{\frac{1}{b^2}(4A^2+b^4)}}$$=\frac{2Ab}{\sqrt{b^4+4A^2}}$