What if you were told that , , and were the midpoints of the sides of and that was the centroid of ? Given the length of , how could you find the lengths of and ? After completing this Concept, you'll be able to use the Median Theorem to solve problems like this one.

### Watch This

First watch this video.

James Sousa: Medians of a Triangle

Now watch this video.

James Sousa: Using the Properties of Medians to Solve for Unknown Values

### Guidance

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

is the median from to the midpoint of .

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 is the centroid, then:

#### Example A

, and are midpoints of the sides of .

a) If , find and .

b) If , find and .

To solve, use the Median Theorem.

a) . .

b)

. is a third of 21, .

#### Example B

is the centroid of and . Find and .

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

#### Example C

, and are the midpoints of each side and is the centroid. If , find and

Use the Median Theorem.

.

Therefore, .

### Guided Practice

1. , and are the midpoints of each side and is the centroid. If , find and

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

3. and are the midpoints of sides and .

a. What is point ?

b. If , find .

c. If and , find and .

**
Answers
**

1. Use the Median Theorem.

.

Therefore,

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

3. Use the Median Theorem.

a. is the centroid.

b. , so .

c.

.

Substitute for to find that .

### Practice

For questions 1-4, , and are the midpoints of each side and is the centroid. Find the following lengths.

- If , find and
- If , find and
- If , find and
- If and , find and .

**
Multistep Problem
**
Find the equation of a median in the
plane.

- Plot and
- Find the midpoint of . Label it .
- Find the slope of .
- Find the equation of .
- Plot
- Find the midpoint of . Label it .
- Find the slope of .
- Find the equation of .

Determine whether the following statement is true or false.

- The centroid is the balancing point of a triangle.