Question

In the given figure, $PQcm,M$ is the mid-point of $QR$?
Also,$MN\perp PR,QS=7cm$ and $TR=21cm$, then $MN=?$

• A. $14cm$
• B. $12.5cm$
• C. $31cm$
• D. $25cm$

Answer 1

The correct option is A $14cm$

$MN\perp PR$

$\angle PMN={90}^{\circ }$

In $\Delta PSQ$ and $\Delta PTR$

$\angle SPQ=\angle TPR$ (common)

$\angle PQS=\angle PRT$ (each ${90}^{\circ }$)

So, $\Delta PSQ\sim \Delta PTR$ (by AA similarity criteria)

$\frac{PQ}{PR}=\frac{SQ}{TR}$ (by C.P.C.T.)

$\frac{24}{PR}=\frac{7}{21}$

$PR=72cm$

$PR=PQ+QR$

$72=24+QR$

$QR=48cm$

Since $M$ is mid-point of $QR$

$QM=MR=\frac{48}{2}=24$

Now, in $\Delta PQS$ and $\Delta PMN$

$\angle PQS-=\angle PMN$ (each ${90}^{\circ }$)

$\angle SPQ=\angle NPM$ (common)

$\Delta PQS\sim \Delta PMN$ (by AA similarity criterion)

$\frac{PQ}{PM}=\frac{SQ}{MN}$

$\frac{24}{PQ+QM}=\frac{7}{MN}$

$\frac{24}{24+24}=\frac{7}{MN}$

$MN=14cm$