# 4.23: Medians

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Line segment that joins a vertex and the midpoint of the opposite side of a triangle.

In a triangle, the line segment that joins a vertex and the midpoint of the opposite side is called a median. Figure $$\PageIndex{1}$$

\overline{LO}\) is the median from L\) to the midpoint of \overline{NM}\).

If you draw all three medians they will intersect at one point called the centroid. Figure $$\PageIndex{2}$$

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. Figure $$\PageIndex{3}$$

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 $$G$$ is the centroid, then:

$$AG=\dfrac{2}{3} AD, CG=\dfrac{2}{3} CF,\:EG=\dfrac{2}{3} BE$$

$$DG=\dfrac{1}{3} AD, FG=\dfrac{1}{3} CF,\:BG=\dfrac{1}{3} BE$$

$$And\: by \:substitution:DG =\dfrac{1}{2} AG,\:FG=\dfrac{1}{2} CG,\:BG=\dfrac{1}{2} EG$$ Figure $$\PageIndex{4}$$

Example $$\PageIndex{1}$$

$$B$$, $$D$$, and $$F$$ are the midpoints of each side and $$G$$ is the centroid. If $$CG=16$$, find $$GF$$ and $$CF$$. Figure $$\PageIndex{5}$$

Solution

Use the Median Theorem.

\begin{align*} CG&=\dfrac{2}{3} CF \\ 16&=\dfrac{2}{3} CF \\ CF&=24.\end{align*}

Therefore, $$GF=8$$.

Example $$\PageIndex{2}$$

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

Solution

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 $$\PageIndex{3}$$

$$I$$, $$K$$, and $$M$$ are midpoints of the sides of $$\Delta HJL$$. Figure $$\PageIndex{6}$$

Solution

If $$JM=18$$, find $$JN$$ and $$NM$$. If $$HN=14$$, find $$NK$$ and $$HK$$.

Use the Median Theorem.

$$JN=\dfrac{2}{3} \cdot 18=12. NM=JM−JN=18−12$$. $$NM=6$$.

$$14=\dfrac{2}{3} \cdot HK$$

$$14\cdot \dfrac{3}{2} =HK=21$$. $$NK$$ is a third of 21, $$NK=7$$.

Example $$\PageIndex{4}$$

H is the centroid of $$\Delta ABC$$ and $$DC=5y−16$$. Find $$x$$ and $$y$$. Figure $$\PageIndex{7}$$

Solution

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

\begin{align*} \dfrac{1}{2} BH=HF &\rightarrow BH=2HF &\qquad HC=\dfrac{2}{3} DC &\rightarrow \dfrac{3}{2} HC=DC \\ 3x+6&=2(2x−1) &\qquad \dfrac{3}{2} (2y+8)&=5y−16\\ 3x+6&=4x−2 &\qquad 3y+12 &=5y−16 \\ 8&=x &\qquad 28&=2y\rightarrow 14=y\end{align*}

Example $$\PageIndex{5}$$

$$B$$, $$D$$, and $$F$$ are the midpoints of each side and G is the centroid. If $$BG=5$$, find $$GE$$ and $$BE$$ Figure $$\PageIndex{8}$$

Solution

Use the Median Theorem.

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

Therefore, $$GE=10$$.

## Review

For questions 1-4, $$B$$, $$D$$, and $$F$$ are the midpoints of each side and $$G$$ is the centroid. Find the following lengths. Figure $$\PageIndex{9}$$
1. If $$CG=16$$, find $$GF$$ and $$CF$$
2. If $$AD=30$$, find $$AG$$ and $$GD$$
3. If $$GF=x$$, find $$GC$$ and $$CF$$
4. If $$AG=9x$$ and $$GD=5x−1$$, find $$x$$ and $$AD$$.

Multi-step Problems Find the equation of a median in the x−y\) plane.

1. Plot $$\Delta ABC:\:A(−6,4)$$,\:B(−2,4)\)\:and\:C(6,−4)\)
2. Find the midpoint of $$\overline{AC}$$. Label it $$D$$.
3. Find the slope of $$\overline{BD}$$.
4. Find the equation of $$\overline{BD}$$.
5. Plot $$\Delta DEF:\: D(−1,5),\:E(1,0),\:F(6,3)$$
6. Find the midpoint of $$\overline{EF}$$. Label it $$G$$.
7. Find the slope of $$\overline{DG}$$.
8. Find the equation of \$$overline{DG}$$.

Determine whether the following statement is true or false.

1. The centroid is the balancing point of a triangle.

## Vocabulary

Term Definition
centroid The centroid is the point of intersection of the medians in a triangle.
Median The median of a triangle is the line segment that connects a vertex to the opposite side's midpoint.

Interactive Element

Video: The Medians of a Triangle

Activities: Medians Discussion Questions

Study Aids: Bisectors, Medians, Altitudes Study Guide

Real World: Medians

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