# 5.1: Midsegments

## Learning Objectives

- Define midsegment.
- Use the Midsegment Theorem.

## Review Queue

Find the midpoint between the given points.

- (-4, 1) and (6, 7)
- (5, -3) and (11, 5)
- Find the equation of the line between (-2, -3) and (-1, 1).
- Find the equation of the line that is parallel to the line from #3 through (2, -7).

**Know What?** A fractal is a repeated design using the same shape (or shapes) of different sizes. Below, is an example of the first few steps of a fractal. Draw the next figure in the pattern.

## Defining Midsegment

**Midsegment:** A line segment that connects two midpoints of the sides of a triangle.

is the midsegment between and .

The tic marks show that and are midpoints.

and

**Example 1:** Draw the midsegment between and for above.

**Solution:** Find the midpoints of and using your ruler. Label these points and . Connect them to create the midsegment.

**Example 2:** You now have all three midpoints of . Draw in midsegment and .

**Solution:**

*For every triangle there are three midsegments.*

## Midsegments in the Plane

Let’s transfer what we know about ** midpoints** in the plane to

**in the plane. We will need to use the midpoint formula, .**

*midsegments*
**Example 3:** 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.

**Solution:** Use the midpoint formula 3 times to find all the midpoints.

and point

and , point

and , point

The graph is to the right.

**Example 4:** Find the slopes of and .

**Solution:** The slope of is . The slope of is .

** From this we can conclude that** . If we were to find the slopes of the other sides and midsegments, we would find and .

**Example 5:** Find and .

**Solution:** Now, we need to find the lengths of and . Use the distance formula.

** From this we can conclude that is half of .** If we were to find the lengths of the other sides and midsegments, we would find that is

**half**of and is

**half**of .

## The Midsegment Theorem

The conclusions drawn in Examples 4 and 5 can be combined into the Midsegment Theorem.

**Midsegment Theorem:** The midsegment of a triangle is half the length of the side it is parallel to.

If is a midsegment of , then and .

**Example 6a:** Mark all the congruent segments on with midpoints , and .

**Solution:** Drawing in all three midsegments, we have:

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

**Example 6b:** Mark all the parallel lines on , with midpoints , and .

**Solution:**

To play with the properties of midsegments, go to http://www.mathopenref.com/trianglemidsegment.html.

**Example 7:** , and are the midpoints of the sides of the triangle.

Find

a)

b)

c) The perimeter of

**Solution:** 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.*

**Example 8:** ** Algebra Connection** Find the value of and . and are midpoints.

**Solution:** . To find , set equal to 17.

**Know What? Revisited** To the left is a picture of the figure in the fractal pattern.

## Review Questions

- Questions 1-5 use the definition of a midsegment and the Midsegment Theorem.
- Questions 6-9 and 18 are similar to Example 7.
- Questions 10-17 are similar to Example 8.
- Questions 19-22 are similar to Example 3.
- Questions 23-30 are similar to Examples 3, 4, and 5.

Determine if 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.
- If a line passes through two sides of a triangle and is parallel to the third side, then it is a midsegment.
- 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 10-17, 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

** Multi-Step Problem** The midpoints of the sides of a triangle are , and . Answer the following questions. The graph is below.

- Find the slope of , and .
- The side that passes through should be parallel to which midsegment? ( are all midsegments of a triangle).
- Using your answer from #24, take the slope of and use the “rise over run” in either direction to create a parallel line to that passes through . Extend it with a ruler.
- Repeat #24 and #25 with and . What are coordinates of the larger triangle?

** Multi-Step Problem** The midpoints of the sides of are , and . Answer the following questions.

- Find the slope of , and .
- Plot the three midpoints and connect them to form midsegment triangle, .
- Using the slopes, find the coordinates of the vertices of . (#22 above)
- Find using the distance formula. Then, find the length of the sides it is parallel to. What should happen?