Question

In the given figure, $\frac{QR}{QS}=\frac{QT}{PR}and$ $\angle 1=\angle 2$. Which of the following is true?

• A. $\Delta PSQ\sim \Delta TQR$
• B. $\Delta QPS\sim \Delta TQR$
• C. $QSP\sim \Delta TQR$
• D. $\Delta PQS\sim \Delta TQR$

Answer 1

The correct option is D $\Delta PQS\sim \Delta TQR$

$In\Delta PQR,\angle PQR=\angle PRQ\left(Given\right)$

$\therefore PQ=PR...\left(i\right)$ (sides opposite to equal angles are equal)

$Given,\frac{QR}{QS}=\frac{QT}{PR}$

Using ($i$), we get
$\frac{QR}{QS}=\frac{QT}{QP}...\left(ii\right)$

In $\Delta PQS$ and $\Delta TQR$,
$\frac{QR}{QS}=\frac{QT}{QP}$ [using ($ii$)]

$\angle Q=\angle Q$ (Common)

$\therefore \Delta PQS\sim \Delta TQR$ (by SAS similarity criterion)