question_answer

PQRS is a rhombus. A straight line through R cuts PS produced at X and PQ produced at Y. If \[SX=\frac{1}{2}PQ\], then the ratio of the length of QY and PQ is

A)  2:1 

B)  1 : 2

C)  \[1:1\]                          

D)  3:1

Correct Answer: A

Solution :

(a): \[PQ=QR=RS=SP\] [PQRS is a rhombus] \[SX=\frac{1}{2}PQ=\frac{1}{2}QR=\frac{1}{2}RS=\frac{1}{2}SP\] In \[\Delta s\] PXY and QRY, \[\angle X=\angle YRQ;\angle P=\angle YQR;\angle Y=\angle Y\therefore \Delta PXY\]and \[\Delta QRY\] are similar. \[\therefore \] \[\frac{PQ+PY}{QY}=\frac{PS+SK}{QR}\] \[\Rightarrow \frac{PQ}{QY}+1=\frac{\frac{3}{2}QR}{QR}=\frac{3}{2}\] \[\Rightarrow \] \[\frac{PQ}{QY}=\frac{3}{2}-1=\frac{1}{2}\Rightarrow \frac{QY}{PQ}=\frac{1}{2}\Rightarrow 2:1\]