1. 5-4 The Triangle Midsegment Theorem Warm Up Lesson Presentation
Lesson Quiz
Holt McDougal Geometry
Holt Geometry

2. Warm Up
Use the points A(2, 2), B(12, 2) and C(4, 8) for Exercises 1–5.
1. Find X and Y, the midpoints of AC and CB.
2. Find XY.
3. Find AB.
4. Find the slope of AB.
5. Find the slope of XY.
6. What is the slope of a line parallel to
3x + 2y = 12?
X (3, 5), Y (8, 5) 5 10

3. Objective
Prove and use properties of triangle midsegments.

4. 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.

5. Every midsegment is parallel and half the size of the side that it is opposite of

6. Steps to confirm if a segment is a midsegment.
Step 1: Find the midpoint of two connecting sides
Step 2: Connect the midpoints and find the slope of it. Find the slope of the third side of the triangle.
Confirm that they are parallel.
Step 3: Find the distance of the two segments from step two. Confirm that the midsegment is ½ of the opposite segment.

7. Example 1: Examining Midsegments in the Coordinate Plane
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.

8. Example 1 Continued
Step 2 Compare the slopes of MN and XY.
Since the slopes are the same,

9. Example 1 Continued
Step 3 Compare the distance of MN and XY.

10. Check It Out! Example 1
The vertices of ΔRST are R(–7, 0), S(–3, 6), and T(9, 2). M is the midpoint of RT, and N is the midpoint of ST. Show that and
Step 1 Find the coordinates of M and N.

11. Check It Out! Example 1 Continued
Step 2 Compare the slopes of MN and RS.
Since the slopes are equal

12. Check It Out! Example 1 Continued
Step 3 Compare the heights of MN and RS.
The length of MN is half the length of RS.

13. Example 2A: Using the Triangle Midsegment Theorem
1. Find each measure. BD
∆ Midsegment Thm.
Substitute 17 for AE.
BD = 8.5
Simplify.

14. Example 2B: Using the Triangle Midsegment Theorem
2. Find each measure.
mCBD
∆ Midsegment Thm.
mCBD = mBDF
Alt. Int. s Thm.
mCBD = 26°
Substitute 26° for mBDF.

15. Check It Out! Example 2a
3. Find each measure. JL
∆ Midsegment Thm.
Substitute 36 for PN and multiply both sides by 2.
2(36) = JL
72 = JL
Simplify.

16. Check It Out! Example 2b
4. Find each measure. PM
∆ Midsegment Thm.
Substitute 97 for LK.
PM = 48.5
Simplify.

17. Check It Out! Example 2c
5. Find each measure.
mMLK
∆ Midsegment Thm.
mMLK = mJMP
Similar triangles
mMLK = 102°
Substitute.

18. 6. In an A-frame support, the distance PQ is 46 inches
6. 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.

19. Lesson Quiz: Part I
Use the diagram for Items 1–3. Find each measure.
7. ED
8. AB
9. mBFE 10 14
44°

20. Lesson Quiz: Part II
10. Find the value of n.
11. ∆XYZ is the midsegment triangle of ∆WUV. What is the perimeter of ∆XYZ? 16
11.5