Question

In the figures, sides XY and YZ and median XA of a triangle XYZ are propotional to sidesDE,EF and median DB of $△DEF$. Show that $\Delta XYZ\sim \Delta DEF$

Answer 1

Consider $\Delta XYA$ and $\Delta DEB$

$\frac{XY}{DE}=\frac{YZ}{EF}=\frac{XA}{DB}$ ..........(1)

$\Rightarrow \Delta XYA\sim \Delta DEB$ .........(By sss)

Since $XA$ and $DB$ are medians,

$2.YA=YZ$..........(2)

$2.EB=EF$ .............(3)

From $\left(1\right),\left(2\right)$ and $\left(3\right)$

$\frac{XY}{DE}=\frac{2.YA}{2.EB}=\frac{XA}{DB}$

Or

$\frac{XY}{DE}=\frac{YZ}{EF}=\frac{XA}{DB}$

Since $\Delta XYA\sim \Delta DEB$, $\angle Y=\angle E$

Thus,

$\Rightarrow \Delta XYZ\sim \Delta DEF$......... (By $SAS$)

Hence proved.