1. Introduction
Triangles are typically thought of as simplistic shapes constructed of three angles and three segments. As we continue to explore this shape, we discover there are many more properties and qualities than we may have first imagined. Each property and quality, such as the midsegment of a triangle, acts as a tool for solving problems.
1.9.3: Proving the Midsegment of a Triangle

2. Key Concepts
The midpoint is the point on a line segment that divides the segment into two equal parts.
A midsegment of a triangle is a line segment that joins the midpoints of two sides of a triangle.
1.9.3: Proving the Midsegment of a Triangle

3. Key Concepts, continued
In the diagram below, the midpoint of is X.
The midpoint of is Y.
A midsegment of is
1.9.3: Proving the Midsegment of a Triangle

4. Key Concepts, continued
The midsegment of a triangle is parallel to the third side of the triangle and is half as long as the third side. This is known as the Triangle Midsegment Theorem.
1.9.3: Proving the Midsegment of a Triangle

5. Key Concepts, continued
Theorem
Triangle Midsegment Theorem
A midsegment of a triangle is parallel to the third side and is half as long.
1.9.3: Proving the Midsegment of a Triangle

6. Key Concepts, continued Every triangle has three midsegments.
1.9.3: Proving the Midsegment of a Triangle

7. Key Concepts, continued
When all three of the midsegments of a triangle are connected, a midsegment triangle is created.
In the diagram below,
1.9.3: Proving the Midsegment of a Triangle

8. Key Concepts, continued
Coordinate proofs, proofs that involve calculations and make reference to the coordinate plane, are often used to prove many theorems.
1.9.3: Proving the Midsegment of a Triangle

9. Common Errors/Misconceptions
assuming a segment that is parallel to the third side of a triangle is a midsegment
incorrectly writing and solving equations to determine lengths
incorrectly calculating slope
incorrectly applying the Triangle Midsegment Theorem to solve problems
misidentifying or leaving out theorems, postulates, or definitions when writing proofs
1.9.3: Proving the Midsegment of a Triangle

10. Guided Practice Example 1
Find the lengths of BC and YZ and the measure of ∠AXZ.
1.9.3: Proving the Midsegment of a Triangle

11. Guided Practice: Example 1, continued Identify the known information.
Tick marks indicate that X is the midpoint of , Y is the midpoint of , and Z is the midpoint of
and are midsegments of
1.9.3: Proving the Midsegment of a Triangle

12. Guided Practice: Example 1, continued Calculate the length of BC.
is the midsegment that is parallel to
The length of is the length of
1.9.3: Proving the Midsegment of a Triangle

13. Guided Practice: Example 1, continued
Triangle Midsegment Theorem
Substitute 4.8 for XZ.
Solve for BC.
1.9.3: Proving the Midsegment of a Triangle

14. Guided Practice: Example 1, continued Calculate the measure of YZ.
is the midsegment parallel to
The length of is the length of
1.9.3: Proving the Midsegment of a Triangle

15. Guided Practice: Example 1, continued
Triangle Midsegment Theorem
Substitute 11.5 for AB.
Solve for YZ.
1.9.3: Proving the Midsegment of a Triangle

16. Guided Practice: Example 1, continued Calculate the measure of .
Triangle Midsegment Theorem
Alternate Interior Angles Theorem
1.9.3: Proving the Midsegment of a Triangle

17. ✔ Guided Practice: Example 1, continued State the answers.
BC is 9.6 units long.
YZ is 5.75 units long.
m∠AXZ is 38°. ✔
1.9.3: Proving the Midsegment of a Triangle

18. Guided Practice: Example 1, continued
1.9.3: Proving the Midsegment of a Triangle

19. Guided Practice Example 3
The midpoints of a triangle are X (–2, 5), Y (3, 1), and Z (4, 8). Find the coordinates of the vertices of the triangle.
1.9.3: Proving the Midsegment of a Triangle

20. Guided Practice: Example 3, continued
Plot the midpoints on a coordinate plane.
1.9.3: Proving the Midsegment of a Triangle

21. Guided Practice: Example 3, continued
Connect the midpoints to form the midsegments , , and
1.9.3: Proving the Midsegment of a Triangle

22. Guided Practice: Example 3, continued
Calculate the slope of each midsegment.
Calculate the slope of
The slope of is
Slope formula
Substitute (–2, 5) and (3, 1) for (x1, y1) and (x2, y2).
Simplify.
1.9.3: Proving the Midsegment of a Triangle

23. Guided Practice: Example 3, continued
Calculate the slope of
The slope of is 7.
Slope formula
Substitute (3, 1) and (4, 8) for (x1, y1) and (x2, y2).
Simplify.
1.9.3: Proving the Midsegment of a Triangle

24. Guided Practice: Example 3, continued
Calculate the slope of
The slope of is .
Slope formula
Substitute (–2, 5) and (4, 8) for (x1, y1) and (x2, y2).
Simplify.
1.9.3: Proving the Midsegment of a Triangle

25. Guided Practice: Example 3, continued
Draw the lines that contain the midpoints.
The endpoints of each midsegment are the midpoints of the larger triangle.
Each midsegment is also parallel to the opposite side.
1.9.3: Proving the Midsegment of a Triangle

26. Guided Practice: Example 3, continued
The slope of is .
From point Y, draw a
line that has a slope of .
1.9.3: Proving the Midsegment of a Triangle

27. Guided Practice: Example 3, continued
The slope of is 7.
From point X, draw a
line that has a slope of 7.
1.9.3: Proving the Midsegment of a Triangle

28. Guided Practice: Example 3, continued
The slope of is
From point Z, draw a
line that has a slope of .
The intersections of the
lines form the vertices
of the triangle.
1.9.3: Proving the Midsegment of a Triangle

29. Guided Practice: Example 3, continued
Determine the vertices of the triangle.
The vertices of the triangle are (–3, –2), (9, 4),
and (–1, 12), as shown on the following slide.
1.9.3: Proving the Midsegment of a Triangle

30. Guided Practice: Example 3, continued✔
1.9.3: Proving the Midsegment of a Triangle

31. Guided Practice: Example 3, continued
1.9.3: Proving the Midsegment of a Triangle