question_answer

X and Y are points on the sides AB and AC respectively of a triangle ABC such that \[\frac{AX}{AB},AY=2\,cm\] and \[YC=6\,cm\]. Find whether \[XY\parallel BC\] or not.

Answer:

\[\frac{AX}{AB}=\frac{1}{4}\]

i.e.,                               \[AX=1K,\text{ }AB=4K\]    (K- constant)

\[\therefore BX=AB-AX\]

\[=4K-1K=3K\]

Now,                 \[\frac{AX}{XB}=\frac{1K}{3K}=\frac{1}{3}\]

And,                 \[\frac{AY}{YC}=\frac{2}{6}=\frac{1}{3}\]

\[\frac{AX}{XB}=\frac{AY}{YC}\]

\[\therefore XY\parallel BC\]         (By converse of Thales? theorem)