Question:
In a right angled triangle with sides a and b and hypotenuse c, the altitude drawn on the hypotenuse is x. Prove that ab = cx.
Solution:
Let $\triangle \mathrm{ABC}$ be a right angle triangle having sides $a$ and $b$; and hypotenuse $c . \mathrm{BD}$ is the altitude drawn on the hypotenuse AC.
Since the altitude is perpendicular on the hypotenuse, both the triangles are similar
$\frac{A B}{B D}=\frac{A C}{B C}$
$\frac{a}{x}=\frac{c}{b}$
$x c=a b$
Hence, $a b=c x$.