In $\triangle PQR$, $S$ is a point on $QR$ such that $\angle Q = \angle SPR$.
- Prove that $\triangle PQR \sim \triangle SPR$
- If $QS = 5 \; cm$, $SR = 4 \; cm$, find the length of $PR$
In triangles $PQR$ and $SPR$,
$\angle Q = \angle SPR$ $\;\;\;$ [given]
$\angle PRQ = \angle PRS$ $\;\;\;$ [common angle]
$\therefore \;$ $\triangle PQR \sim \triangle SPR$ $\;\;\;$ [by angle-angle (AA) postulate]
$\because \;$ Corresponding sides of similar triangles are in proportion, we have,
$\dfrac{QR}{PR} = \dfrac{PR}{SR}$
$\implies$ $PR^2 = QR \times SR$
i.e. $\;$ $PR^2 = \left(QS + SR\right) = \left(5 + 4\right) \times 4 = 36$
$\implies$ $PR = 6 \; cm$