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 .
Explore More
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 915, 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