Question

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

• A. $\frac{Ab}{\sqrt{b^4+4A^2}}$
• B. $\frac{3Ab}{\sqrt{b^2+4A^2}}$
• C. $\frac{2Ab}{\sqrt{b^4+4A^2}}$
• D. None of these

Answer 1

Answer: (C)

$\frac{2Ab}{\sqrt{b^4+4A^2}}$

Let AC $=$ b and AD $=$ h.
Then, $DC=\sqrt{b^2-h^2}$

Also, $\Delta ABC\sim \Delta DAC$
$\frac{AB}{AC}=\frac{DA}{DC}\Rightarrow \frac{AB}{b}=\frac{h}{\sqrt{b^2-h^2}}$

Also, $A=\frac{1}{2}.AC.AB=\frac{1}{2}.b.\frac{bh}{\sqrt{b^2-h^2}}$
$\Rightarrow h=\frac{2Ab}{\sqrt{b^2+4A^2}}$