Question

A right triangle has a hypotenuse of length $pcm$ and one side of length $qcm$. If $p-q=1$, find the length of the third side of the triangle :

Answer 1

Pythagoras theorem:

Given hypotenuse $=pcm$

Length of another side $=qcm$

Let $x$ be the length of the other side.

Given that $p-q=1$

$p=q+1$

By Pythagoras theorem

$$\begin{gathered} p^2=q^2+x^2 \\ {x}^2=p^2-q^2 \\ =\left(p-q\right)\left(p+q\right) \\ =\left(1\right)\left(p+q\right) \\ =\left(q+q+1\right) \\ {x}^2=\left(2q+1\right) \\ x=\sqrt{2q+1} \end{gathered}$$

Hence, the length of the third side of the triangle is $\sqrt{2q+1}$.