Midpoints of a Triangle

In this video, we are going to look at the relationships that exist among line segments when the midpoints of a triangle are connected.

For example:
Given triangle ABC, each line segment has a midpoint. Let’s label each midpoint as “X”, “Y”, and “Z”. If we connect the midpoints, we can see that we have another triangle. When we draw in midpoints, we are given two new line segments that are congruent to each other. Therefore:
\overline{BX} \cong \overline{XC}
\overline{AY} \cong \overline{YC}
\overline{AZ} \cong \overline{ZB}

Now, let’s look at the new triangle that we drew, triangle XYZ. We can see that:
\overline{XZ} \| \overline{AC}
\overline{XY} \| \overline{AB}
\overline{YZ} \| \overline{BC}

We also have some parallelograms, like AZXY and BXYZ. We know that in parallelograms, opposite sides are parallel, and also congruent. Therefore:
\overline{AY} \cong \overline{ZX}
\overline{AZ} \cong \overline{YX}
\overline{BX} \cong \overline{ZY}

So to sum it all up:
\overline{XZ} \| \overline{AC}
\overline{XY} \| \overline{AB}
\overline{YZ} \| \overline{BC}
\overline{XY}=\frac{1}{2}\overline{AB}
\overline{YZ}=\frac{1}{2}\overline{BC}
\overline{XZ}=\frac{1}{2}\overline{AC}
P_{\triangle{XYZ}}=\frac{1}{2}P_{\triangle{ABC}}