5.1: Midsegment Theorem
What if you created a repeated design using the same shape (or shapes) of different sizes? This would be called a fractal. Below, is an example of the first few steps of one. What does the next figure look like? How many triangles are in each figure (green and white triangles)? Is there a pattern? After completing this Concept, you'll be able to better understand how these fractals are created.
Watch This
CK12 Foundation: Chapter5MidsegmentTheoremA
James Sousa: Introduction to the Midsegments of a Triangle
James Sousa: Determining Unknown Values Using Properties of the Midsegments of a Triangle
Guidance
A midsegment is a line segment that connects two midpoints of adjacent sides of a triangle. For every triangle there are three midsegments. The Midsegment Theorem states that the midsegment of a triangle is half the length of the side it is parallel to.
Example A
Draw the midsegment
Find the midpoints of
Don’t forget to put the tic marks, indicating that
Example B
Find the midpoint of
Example C
Mark everything you have learned from the Midsegment Theorem on
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
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
To play with the properties of midsegments, go to http://www.mathopenref.com/trianglemidsegment.html.
Example D
Find
a)
b)
c) The perimeter of
Use the Midsegment Theorem.
a)
b)
c) The perimeter is the sum of the three sides of
Watch this video for help with the Examples above.
CK12 Foundation: Chapter5MidsegmentTheoremB
Concept Problem Revisited
To the left is a picture of the
Vocabulary
A line segment that connects two midpoints of the sides of a triangle is called a midsegment. A midpoint is a point that divides a segment into two equal pieces. Two lines are parallel if they never intersect. Parallel lines have slopes that are equal. In a triangle, midsegments are always parallel to one side of the triangle.
Guided Practice
The vertices of
1. Find the midpoints of all three sides, label them
2. Find the slopes of
3. Find
4. If the midpoints of the sides of a triangle are
Answers:
1. Use the midpoint formula 3 times to find all the midpoints.
The graph would look like the graph below.
2. The slope of
The slope of
From this we can conclude that
3. Now, we need to find the lengths of
Note that
4. 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
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).
Practice
 If
OP=12 , findRS andTU .  If
RS=8 , findTU .  If
RS=2x , andOP=20 , findx andTU .  If
OP=4x andRS=6x−8 , findx .  Is
△XOP≅△YOP ? Why or why not?
For questions 613, find the indicated variable(s). You may assume that all line segments within a triangle are midsegments.
 The sides of
△XYZ are 26, 38, and 42.△ABC is formed by joining the midpoints of△XYZ . Find the perimeter of
△ABC .  Find the perimeter of
△XYZ .  What is the relationship between the perimeter of a triangle and the perimeter of the triangle formed by connecting its midpoints?
 Find the perimeter of
Coordinate Geometry Given the vertices of

A(5,−2),B(9,4) andC(−3,8) 
A(−10,1),B(4,11) andC(0,−7) 
A(0,5),B(4,−1) andC(−2,−3) 
A(2,4),B(8,−4) andC(2,−4)
For questions 1922,
 Find the midpoints of sides
CA¯¯¯¯¯ andAT¯¯¯¯¯ . Label themL andM respectively.  Find the slopes of
LM¯¯¯¯¯¯ andCT¯¯¯¯¯ .  Find the lengths of
LM¯¯¯¯¯¯ andCT¯¯¯¯¯ .  What have you just proven algebraically?
Congruent
Congruent figures are identical in size, shape and measure.Midpoint Formula
The midpoint formula says that for endpoints and , the midpoint is .midsegment
A midsegment connects the midpoints of two sides of a triangle or the nonparallel sides of a trapezoid.Image Attributions
Description
Learning Objectives
Here you'll learn what a midsegment is, and how midsegments in triangles relate to the triangles.