# Triangle Midsegment Theorem

## Midsegment

A midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle.

In the figure $D$ is the midpoint of $\overline{AB}$ and $E$ is the midpoint of $\overline{AC}$.

So, $\overline{DE}$ is a midsegment.

## The Triangle Midsegment Theorem

A midsegment connecting two sides of a triangle is parallel to the third side and is half as long.

If $AD=DB\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}AE=EC,$

then $\overline{DE}\parallel \overline{BC}$ and $DE=\frac{1}{2}BC$ .

Example :

Find the value of $x$.

Here $P$ is the midpoint of $AB$, and $Q$ is the midpoint of $BC$. So, $\overline{PQ}$ is a midsegment.

Therefore by the Triangle Midsegment Theorem,

$PQ=\frac{1}{2}BC$

Substitute.

$\begin{array}{l}x=\frac{1}{2}\cdot 6\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=3\end{array}$

The value of $x$ is $3$.