Question

In the given figure, PQ || TS || MR. If TN = 6 cm, NR = 12 cm, PQ = 12 cm and MR = 18 cm, then the length of PR is

• A. 24 cm
• B. 20 cm
• C. 25 cm
• D. 27 cm

Answer 1

The correct option is A 24 cm

Given: PQ||TS||MR, TN = 6 cm, NR = 12 cm, PQ = 12 cm, MR = 18 cm


In $\Delta TNS$ and $\Delta RNM$,
$\angle TNS=\angle MNR$ (Vertically opposite angle)
$\angle TSN=\angle NMR$ (Alternate interior angle)
$\therefore \Delta TNS\sim \Delta RNM$ (AA similarity criteria)
Thus, $\frac{TN}{RN}=\frac{TS}{MR}$
$\Rightarrow \frac{6}{12}=\frac{TS}{18}$
$\Rightarrow TS=9cm$
Now, in ΔPQR, PQ || TS
Also, using BPT, $\frac{PT}{PR}=\frac{QS}{QR}$
$\angle RTS=\angle RPQ$ (Corresponding angles)
And, $\angle PRQ=\angle TRS$ (Common)
Thus, $\Delta PQR\sim \Delta TSR$ (AA similarity criteria)
$\therefore \frac{PQ}{TS}=\frac{PR}{TR}$
$\Rightarrow \frac{12}{9}=\frac{PR}{18}$ $(TR=TN+NR$)
$\Rightarrow PR=\frac{12}{9}\times 18$
$\Rightarrow PR=24cm$
Hence, the correct answer is option (1).