1. A midsegment of a triangle is a segment that joins the midpoints of two sides of the triangle. Every triangle has three midsegments, which form the midsegment triangle.

2. Since the slopes are the same,
Example 1:
The vertices of ∆XYZ are X(–1, 8), Y(9, 2), and Z(3, –4). M and N are the midpoints of XZ and YZ. Show that and
Step 1 Find the coordinates of M and N.
Step 2 Compare the slopes of MN and XY.
Since the slopes are the same,

3. Step 3 Compare the heights of MN and XY.

4. The relationship shown in Example 1 is true for the three midsegments of every triangle.

5. Example 2A:
Find each measure. BD
∆ Midsegment Thm.
Substitute 17 for AE.
BD = 8.5
Simplify.
mCBD
∆ Midsegment Thm.
mCBD = mBDF
Alt. Int. s Thm.
mCBD = 26°
Substitute 26° for mBDF.

6. Geometry 5.4 Triangle Midsegment Theorem
Example 3: Indirect Measurement Application
In an A-frame support, the distance PQ is 46 inches. What is the length of the support ST if S and T are at the midpoints of the sides?
∆ Midsegment Thm.
Substitute 46 for PQ.
ST = 23
Simplify.
The length of the support ST is 23 inches.