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Medians

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

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Practice Medians
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Medians

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.

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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, .

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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 .

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 .

Explore More

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

1. If , find and
2. If , find and
3. If , find and
4. If and , find and .

Multi-step Problems Find the equation of a median in the plane.

1. Plot and
2. Find the midpoint of . Label it .
3. Find the slope of .
4. Find the equation of .
5. Plot
6. Find the midpoint of . Label it .
7. Find the slope of .
8. Find the equation of .

Determine whether the following statement is true or false.

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

Vocabulary Language: English

centroid

centroid

The centroid is the point of intersection of the medians in a triangle.
Median

Median

The median of a triangle is the line segment that connects a vertex to the opposite side's midpoint.