Question

Ray PT is the bisector of $\angle QPR$. Find the value of x and perimeter of $△\mathrm{PQR}$

Answer 1

Given: Ray PT is the bisector of $\angle \mathrm{QPR}$.

To find: x and the perimeter of $△\mathrm{PQR}$

In a triangle, the angle bisector divides the side opposite to the angle in the ratio of the remaining sides.

$$\begin{gathered} \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ \frac{\mathrm{QT}}{\mathrm{TR}}=\frac{\mathrm{QP}}{\mathrm{RP}} \\ \Rightarrow \frac{4}{5}=\frac{3.6}{x} \\ \Rightarrow x=\frac{3.6\times 5}{4} \\ \Rightarrow x=4.5\mathrm{cm} \\ \mathrm{Perimeter}\mathrm{of}△\mathrm{PQR}=\mathrm{PQ}+\mathrm{QR}+\mathrm{RT}=\mathrm{PQ}+\mathrm{QT}+\mathrm{TR}+\mathrm{PR} \\ =(3.6+4+5+4.5)\mathrm{cm}=17.1\mathrm{cm} \end{gathered}$$