What if you were given and told that was its midsegment? How could you find the length of given the length of the triangle's third side, ? After completing this Concept, you'll be able to use the Midsegment Theorem to solve problems like this one.

### Watch This

First watch this video.

James Sousa: Introduction to the Midsegments of a Triangle

Now watch this video.

James Sousa: Determining Unknown Values Using Properties of the Midsegments of a Triangle

### Guidance

A line segment that connects two midpoints of the sides of a triangle is called a **midsegment**. is the midsegment between and .

The tic marks show that and are midpoints. and . For every triangle there are three midsegments.

There are two important properties of midsegments that combine to make the **Midsegment Theorem**. The **Midsegment Theorem** states that the midsegment connecting the midpoints of two sides of a triangle is parallel to the third side of the triangle, and the length of this midsegment is half the length of the third side. So, if is a midsegment of , then and .

Note that there are two important ideas here. One is that the midsegment is parallel to a side of the triangle. The other is that the midsegment is always half the length of this side. To play with the properties of midsegments, go to http://www.mathopenref.com/trianglemidsegment.html.

#### Example A

The vertices of are and . Find the midpoints of all three sides, label them and Then, graph the triangle, plot the midpoints and draw the midsegments.

To solve this problem, use the midpoint formula 3 times to find all the midpoints. Recall that the midpoint formula is .

and point

and , point

and , point

The graph is to the right.

#### Example B

Mark all the congruent segments on with midpoints , and .

Drawing in all three midsegments, we have:

Also, this means the four smaller triangles are congruent by SSS.

Now, mark all the parallel lines on , with midpoints , and .

#### Example C

, and are the midpoints of the sides of .

Find

a)

b)

c) The perimeter of

To solve, use the Midsegment Theorem.

a)

b)

c) Add up the three sides of to find the perimeter.

*Remember: No line segment over means length or distance.*

### Guided Practice

1. Find the value of and . and are midpoints.

2. True or false: If a line passes through two sides of a triangle and is parallel to the third side, then it is a midsegment.

3. Find . You may assume that the line segment within the triangle is a midsegment.

**Answers:**

1. . To find , set equal to 17.

2. This statement is false. A line that passes through two sides of a triangle is only a midsegment if it passes through the **midpoints** of the two sides of the triangle.

3. Because a midsegment is always half the length of the side it is parallel to, we know that .

### Practice

Determine whether each statement is true or false.

- The endpoints of a midsegment are midpoints.
- A midsegment is parallel to the side of the triangle that it does not intersect.
- There are three congruent triangles formed by the midsegments and sides of a triangle.
- There are three midsegments in every triangle.

, and are midpoints of the sides of and .

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

For questions 9-15, find the indicated variable(s). You may assume that all line segments within a triangle are midsegments.

- The sides of are 26, 38, and 42. is formed by joining the midpoints of .
- What are the lengths of the sides of ?
- Find the perimeter of .
- Find the perimeter of .
- What is the relationship between the perimeter of a triangle and the perimeter of the triangle formed by connecting its midpoints?

** Coordinate Geometry** Given the vertices of below find the midpoints of each side.

- and
- and
- and
- and