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 :

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Solution

## 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${p}^{2}={q}^{2}+{x}^{2}\phantom{\rule{0ex}{0ex}}{x}^{2}={p}^{2}-{q}^{2}\phantom{\rule{0ex}{0ex}}=\left(p-q\right)\left(p+q\right)\phantom{\rule{0ex}{0ex}}=\left(1\right)\left(p+q\right)\phantom{\rule{0ex}{0ex}}=\left(q+q+1\right)\phantom{\rule{0ex}{0ex}}{x}^{2}=\left(2q+1\right)\phantom{\rule{0ex}{0ex}}x=\sqrt{2q+1}$Hence, the length of the third side of the triangle is $\sqrt{2q+1}$.

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