Question

$△\mathrm{ABC}$ is a triangle right angled at $\mathrm{C}$. A line through the mid-point $\mathrm{M}$ of hypotenuse $\mathrm{AB}$ and parallel to$\mathrm{BC}$ intersects $\mathrm{AC}$at $D$. Show that $D$ is the mid-point of$\mathrm{AC}$

Answer 1

Showing that D is the mid-point of AC:

Given: $\mathrm{M}$ is the midpoint of $\mathrm{AB}$ and $\mathrm{MD}\parallel \mathrm{BC}$

Considering $∆ABC$

We observe that $\mathrm{M}$ is the midpoint of side $AB$ and $\mathrm{DM}\left|\right|\mathrm{BC}$

This implies, $\mathrm{DC}=\mathrm{AD}$ (Converse of mid-point theorem)

Hence, $\mathrm{D}$ is the mid-point of $\mathrm{AC}$.