# 5.1: Midsegments of a Triangle

## Learning Objectives

- Identify the midsegments of a triangle.
- Use the Midsegment Theorem to solve problems involving side lengths, midsegments, and algebra.

## Review Queue

Find the midpoint between the given points.

- (-4, 1) and (6, 7)
- (5, -3) and (11, 5)
- (0, -2) and (-4, 14)
- 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 #4 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. What does the next figure look like? How many triangles are in each figure (green and white triangles)? Is there a pattern?

## Defining Midsegment

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

**Example 1:** Draw the midsegment between and . Use appropriate tic marks.

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

Don’t forget to put the tic marks, indicating that and are midpoints, and .

**Example 2:** Find the midpoint of from . Label it and find the other two midsegments of the triangle.

**Solution:**

*For every triangle there are three midsegments.*

Let’s transfer what we know about midpoints in the coordinate plane to midsegments in the coordinate plane. We will need to use the midpoint formula, .

**Example 3:** The vertices of are and . Find the midpoints of all three sides, label them and . Then, graph the triangle, it’s midpoints and draw in the midsegments.

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

and , point

and , point

and , point

The graph would look like the graph to the right. We will use this graph to explore the properties of midsegments.

**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 . *This is a property of all midsegments.*

**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 . *This is a property of all midsegments.*

## The Midsegment Theorem

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

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

**Example 6:** Mark everything you have learned from the Midsegment Theorem on above.

**Solution:** Let’s draw two different triangles, one for the congruent sides, and one for the parallel lines.

Because the midsegments are half the length of the sides they are parallel to, they are congruent to half of each of those sides (as marked). Also, this means that all four of the triangles in , created by the midsegments are congruent by SSS.

As for the parallel midsegments and sides, several congruent angles are formed. In the picture to the right, the pink and teal angles are congruent because they are corresponding or alternate interior angles. Then, the purple angles are congruent by the Angle Theorem.

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) The perimeter is the sum of the three sides of .

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

**Solution:** First, is half of 34, or 17. To find , set equal to 17.

Let’s go back to the coordinate plane.

**Example 9:** If the midpoints of the sides of a triangle are , and , find the vertices of the triangle.

**Solution:** The easiest way to solve this problem is to graph the midpoints and then apply what we know from the Midpoint Theorem.

Now that the points are plotted, find the slopes between all three.

slope

slope

slope

Using the slope between two of the points and the third, plot the slope triangle on either side of the third point and extend the line. Repeat this process for all three midpoints. For example, use the slope of with point .

The green lines in the graph to the left represent the slope triangles of each midsegment. The three dotted lines represent the sides of the triangle. Where they intersect are the vertices of the triangle (the blue points), which are (-8, 8), (10, 2) and (-2, 6).

**Know What? Revisited** To the left is a picture of the figure in the fractal pattern. The number of triangles in each figure is 1, 4, 13, and 40. The pattern is that each term increase by the next power of 3.

## Review Questions

and are midpoints of the sides of and .

- If , find and .
- If , find .
- If , and , find and .
- If and , find .
- Is ? Why or why not?

For questions 6-13, 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 .
- 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 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 .
- Find using the distance formula. Then, find the length of the sides it is parallel to. What should happen?

** More Coordinate Geometry** Given the midpoints of the sides of a triangle, find the vertices of the triangle. Refer back to problems 19-21 for help.

- (-2, 1), (0, -1) and (-2, -3)
- (1, 4), (4, 1) and (2, 1)

Given the vertices of , find:

a) the midpoints of and where is the midpoint of , is the midpoint of and is the midpoint of .

b) Show that midsegments and are parallel to sides and respectively.

c) Show that midsegments and are half the length of sides and respectively.

- and
- and

For questions 27-30, has vertices and .

- Find the midpoints of sides and . Label them and respectively.
- Find the slopes of and .
- Find the lengths of and .
- What have you just proven algebraically?