Chapter 8.1, Problem 42WE

To determine

To prove: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

Explanation of Solution

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,

  $\Delta CBD\sim \Delta CAB$

If triangles are similar then the corresponding sides are proportional.

Hence,

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

So,

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