Question : $\triangle \mathrm{XYZ} \sim \triangle \mathrm{GST}$ and $\mathrm{XY}: \mathrm{GS}=2: 3, \mathrm{XV}$ is the median to the side $\mathrm{YZ}$, and $\mathrm{GD}$ is the median to the side ST. The value of $\left(\frac{\mathrm{YV}}{\mathrm{SD}}\right)^2$ is:
Option 1: $\frac{4}{9}$
Option 2: $\frac{3}{5}$
Option 3: $\frac{1}{4}$
Option 4: $\frac{2}{3}$
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Correct Answer: $\frac{4}{9}$
Solution :
Given: $\Delta \mathrm{XYZ} \sim \Delta \mathrm{GST}$
$\left(\frac{\mathrm{YV}}{\mathrm{SD}}\right)=\left(\frac{\mathrm{XY}}{\mathrm{GS}}\right)$
$\therefore\left(\frac{\mathrm{YV}}{\mathrm{SD}}\right)^2=\left(\frac{\mathrm{XY}}{\mathrm{GS}}\right)^2=\left(\frac{\mathrm{2}}{\mathrm{3}}\right)^2=\frac{4}{9}$
Hence, the correct answer is $\frac{4}{9}$.
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