Question

Prove: In a right triangle, the product of the hypotenuse and the length of the altitude drawn to the hypotenuse equals the product of the two legs.

Short answer

The product of the two legs is equal to the product of the hypotenuse and the length of the altitude drawn to the hypotenuse in a right triangle.

Step-by-step solution

Step 1. Given information.

A right angle triangle is given,

In which an altitude is drawn to the hypotenuse.

Step 2. Proof.

Consider the following figure,

Here,

$ABC$ is a right-angled triangle such that,

$\angle B=90°$

Also,

$BD\perp AC$

If in a right-angled triangle, a perpendicular is drawn from the right angle to the base, the triangles adjoining the perpendicular are similar both to the whole and to each other.

So,

$\Delta CBD~\Delta CAB$

Step 3. Concept used.

If triangles are similar then the corresponding sides are proportional.

Hence,

$\begin{array}{l}\frac{CB}{CA}=\frac{BD}{AB}\\ CB\times AB=BD\times CA\end{array}$

Therefore,

In a right triangle, the product of the hypotenuse and the length of the altitude drawn to the hypotenuse equals the product of the two legs.