### Medians

In a triangle, the line segment that joins a vertex and the midpoint of the opposite side is called a **median**.

If you draw all three medians they will intersect at one point called the **centroid**.

The centroid is the “balancing point” of a triangle. This means that if you were to cut out the triangle, the centroid is its center of gravity so you could balance it there.

The **Median Theorem** states that the medians of a triangle intersect at a point called the centroid that is two-thirds of the distance from the vertices to the midpoint of the opposite sides.

So if

What if you were told that

### Examples

#### Example 1

Use the Median Theorem.

Therefore,

#### Example 2

True or false: The median bisects the side it intersects.

This statement is true. By definition, a median intersects a side of a triangle at its midpoint. Midpoints divide segments into two equal parts.

#### Example 3

If

Use the Median Theorem.

#### Example 4

To solve, use the Median Theorem. Set up and solve equations.

#### Example 5

Use the Median Theorem.

\begin{align*}BG&=\frac{1}{3}BE \\ 5 &=\frac{1}{3}BE \\ BE &=15\\\end{align*}.

Therefore, \begin{align*}GE=10\end{align*}.

### Review

For questions 1-4, \begin{align*}B, \ D\end{align*}, and \begin{align*}F\end{align*} are the midpoints of each side and \begin{align*}G\end{align*} is the centroid. Find the following lengths.

- If \begin{align*}CG = 16\end{align*}, find \begin{align*}GF\end{align*} and \begin{align*}CF\end{align*}
- If \begin{align*}AD = 30\end{align*}, find \begin{align*}AG\end{align*} and \begin{align*}GD\end{align*}
- If \begin{align*}GF = x\end{align*}, find \begin{align*}GC\end{align*} and \begin{align*}CF\end{align*}
- If \begin{align*}AG = 9x\end{align*} and \begin{align*}GD = 5x - 1\end{align*}, find \begin{align*}x\end{align*} and \begin{align*}AD\end{align*}.

** Multi-step Problems** Find the equation of a median in the \begin{align*}x-y\end{align*} plane.

- Plot \begin{align*}\triangle ABC: \ A(-6, 4), \ B(-2, 4)\end{align*} and \begin{align*}C(6, -4)\end{align*}
- Find the midpoint of \begin{align*}\overline{AC}\end{align*}. Label it \begin{align*}D\end{align*}.
- Find the slope of \begin{align*}\overline{BD}\end{align*}.
- Find the equation of \begin{align*}\overline{BD}\end{align*}.
- Plot \begin{align*}\triangle DEF: \ D(-1, 5), \ E(1, 0), \ F(6, 3)\end{align*}
- Find the midpoint of \begin{align*}\overline{EF}\end{align*}. Label it \begin{align*}G\end{align*}.
- Find the slope of \begin{align*}\overline{DG}\end{align*}.
- Find the equation of \begin{align*}\overline{DG}\end{align*}.

Determine whether the following statement is true or false.

- The centroid is the balancing point of a triangle.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.4.