question_answer

If in a triangles XYZ, P, Q are points on XY, YZ respectively such that XP = 2PY, XQ = 2QZ, then the ratio, area of\[\phi \]area of \[x,y,z\], is

A)  4 : 9                 

B)         2 : 3

C)  3 : 2               

D)         9 : 4

Correct Answer: A

Solution :

 Given   \[\frac{XP}{XY}=\frac{2}{3}\]  Referring figure,                 \[\frac{Area\,\,of\,\,\Delta \,XPQ}{Area\,\,of\,\,\Delta \,XYZ}=\frac{\frac{1}{2}PQ\times XL}{\frac{1}{2}\times YZ\times XM}\]                 \[=\frac{PQ}{YZ}\times \frac{PQ}{YZ}\]                 \[{{\left( \frac{PQ}{YZ} \right)}^{2}}={{\left( \frac{2}{3} \right)}^{2}}=\frac{4}{9}\]