Question

A right-angle triangle has a hypotenuse of length $Q\mathrm{cm}$ and one side of length $P\mathrm{cm}$. If $Q-P=2$ express the length of the third side of the right-angle triangle in terms of $Q$.

Answer 1

Express the third side in terms of $Q$.

According to the Pythagoras theorem, In a right-angle triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides.

Here, $\mathrm{P}\mathrm{cm}$ is the length of the hypotenuse and $Q$ is one of the other sides of right angle triangle.

Assume $x$ the the third side of the triangle.

Find the length of the third side of the right-angle triangle in terms of $Q$:

$$\begin{gathered} {\mathrm{Q}}^2={\mathrm{P}}^2+{\mathrm{x}}^2\left[\mathrm{by}\mathrm{pythagoras}\mathrm{theorem}\right] \\ \Rightarrow x^2={\mathrm{Q}}^2-{\mathrm{P}}^2 \\ \Rightarrow x^2=(\mathrm{Q}-\mathrm{P})(\mathrm{Q}+\mathrm{P}) \\ \Rightarrow x^2=2(\mathrm{Q}+\mathrm{P})\mathbf{[}\mathbf{\because }\mathbf{}\mathit{Q}\mathbf{-}\mathit{P}\mathbf{=}\mathbf{2}\mathit{]} \\ \mathit{\Rightarrow }\mathit{}\mathit{}\mathit{}\mathit{}{x}^{\mathit{2}}\mathit{=}2(\mathrm{Q}+\mathrm{Q}-2) \\ \Rightarrow x^2=4\mathrm{Q}-4 \\ \Rightarrow x^2=4(\mathrm{Q}-1) \\ \Rightarrow x=\sqrt{4(\mathrm{Q}-1)} \\ \Rightarrow x=2\sqrt{\mathrm{Q}-1} \end{gathered}$$

Hence, the third side length is $2\sqrt{Q-1}$.