Question

The sides of triangle ABC measure 5, 6 and 7 units, while the perimeter of $△\mathrm{PQR}$ is 360 units .
If $△\mathrm{ABC}$ is similar to $△\mathrm{PQR}$, then find the sides of $△\mathrm{PQR}$.

Answer 1

We have to find the sides of $△\mathrm{PQR}$
It is given that $△\mathrm{ABC}~△\mathrm{PQR}$.
$$\begin{gathered} \mathrm{Therefore},\mathrm{by}\mathrm{the}\mathrm{property}\mathrm{of}\mathrm{similarity}, \\ \frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\mathrm{BC}}{\mathrm{QR}}=\frac{\mathrm{AC}}{\mathrm{PR}} \\ \mathrm{This}\mathrm{is}\mathrm{also}\mathrm{equal}\mathrm{to}\frac{\mathrm{AB}+\mathrm{BC}+\mathrm{AC}}{\mathrm{PQ}+\mathrm{QR}+\mathrm{PR}}. \\ \frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\mathrm{BC}}{\mathrm{QR}}=\frac{\mathrm{AC}}{\mathrm{PR}}=\frac{\mathrm{AB}+\mathrm{BC}+\mathrm{AC}}{\mathrm{PQ}+\mathrm{QR}+\mathrm{PR}} \\ \therefore \frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\mathrm{AB}+\mathrm{BC}+\mathrm{AC}}{\mathrm{PQ}+\mathrm{QR}+\mathrm{PR}} \\ \Rightarrow \frac{5}{\mathrm{PQ}}=\frac{5+6+7}{360} \\ \Rightarrow \mathrm{PQ}=\frac{360\times 5}{18} \\ \Rightarrow \mathrm{PQ}=100\mathrm{units}. \\ \therefore \frac{\mathrm{BC}}{\mathrm{QR}}=\frac{\mathrm{AB}+\mathrm{BC}+\mathrm{AC}}{\mathrm{PQ}+\mathrm{QR}+\mathrm{PR}} \\ \Rightarrow \frac{6}{\mathrm{QR}}=\frac{18}{360} \\ \Rightarrow \mathrm{QR}=\frac{360\times 6}{18} \\ \Rightarrow \mathrm{QR}=120\mathrm{units}. \\ \therefore \frac{\mathrm{AC}}{\mathrm{PR}}=\frac{\mathrm{AB}+\mathrm{BC}+\mathrm{AC}}{\mathrm{PQ}+\mathrm{QR}+\mathrm{PR}} \\ \Rightarrow \frac{7}{\mathrm{PR}}=\frac{18}{360} \end{gathered}$$

$$\begin{gathered} \Rightarrow PR=\frac{360\times 7}{18} \\ \Rightarrow PR=140\mathrm{units} \end{gathered}$$

Thus, the sides are 100, 120 and 140 units.