Question

In triangle $PQR$ right angled at $Q$. If $\mathrm{PR}+\mathrm{QR}=25\mathrm{cm}$ and $PQ=5cm$ find all other sides.

Answer 1

Find the remaining sides of the given triangle.

In the question, it is given that in triangle $PQR$ right angled at $Q$.

If $\mathrm{PR}+\mathrm{QR}=25\mathrm{cm}\left(i\right)$ and $PQ=5cm$.

$$\begin{gathered} {\mathrm{PR}}^2={\mathrm{PQ}}^2+{\mathrm{QR}}^2\left[\because \mathrm{PQ}=5\mathrm{cm}\mathrm{given}\right] \\ \Rightarrow 25={\mathrm{PR}}^2-{\mathrm{QR}}^2\left[\because {\mathrm{a}}^2-{\mathrm{b}}^2=\left(\mathrm{a}+\mathrm{b}\right)\left(\mathrm{a}-\mathrm{b}\right)\right] \\ \Rightarrow 25=\left(\mathrm{PR}+\mathrm{QR}\right)\left(\mathrm{PR}-\mathrm{QR}\right)\left[\because \mathrm{PR}+\mathrm{QR}=25\mathrm{cm}\right] \\ \Rightarrow \mathrm{PR}-\mathrm{QR}=1\mathrm{cm}\dots \left(ii\right) \end{gathered}$$

Add equation $\left(i\right)$ and equation $\left(ii\right)$.

$$\begin{gathered} 2\mathrm{PR}=26\mathrm{cm} \\ \Rightarrow \mathrm{PR}=13\mathrm{cm} \end{gathered}$$

Subtract equation $\left(ii\right)$ from equation $\left(i\right)$.

$$\begin{gathered} 2\mathrm{QR}=24\mathrm{cm} \\ \Rightarrow \mathrm{QR}=12\mathrm{cm} \end{gathered}$$

Hence $\mathrm{PQ}=5\mathrm{cm},\mathrm{QR}=12\mathrm{cm},\mathrm{PR}=13\mathrm{cm}$.