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 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. 2 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. 3 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. 4 1.9.3: Proving the Midsegment of a Triangle

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

6. Key Concepts, continued Every triangle has three midsegments. 6 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,. 7 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. 8 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 9 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. 10 1.9.3: Proving the Midsegment of a Triangle

11. Guided Practice: Example 1, continued 1.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 11 1.9.3: Proving the Midsegment of a Triangle

12. Guided Practice: Example 1, continued 2.Calculate the length of BC. is the midsegment that is parallel to. The length of is the length of. 12 1.9.3: Proving the Midsegment of a Triangle

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

14. Guided Practice: Example 1, continued 3.Calculate the measure of YZ. is the midsegment parallel to. The length of is the length of. 14 1.9.3: Proving the Midsegment of a Triangle

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

16. Guided Practice: Example 1, continued 4.Calculate the measure of. 16 1.9.3: Proving the Midsegment of a Triangle Triangle Midsegment Theorem Alternate Interior Angles Theorem

17. Guided Practice: Example 1, continued 5.State the answers. BC is 9.6 units long. YZ is 5.75 units long. m ∠ AXZ is 38°. 17 1.9.3: Proving the Midsegment of a Triangle ✔

18. Guided Practice: Example 1, continued 18 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. 19 1.9.3: Proving the Midsegment of a Triangle

20. Guided Practice: Example 3, continued 1.Plot the midpoints on a coordinate plane. 20 1.9.3: Proving the Midsegment of a Triangle

21. Guided Practice: Example 3, continued 2.Connect the midpoints to form the midsegments,, and. 21 1.9.3: Proving the Midsegment of a Triangle

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

23. Guided Practice: Example 3, continued Calculate the slope of. The slope of is 7. 23 1.9.3: Proving the Midsegment of a Triangle Slope formula Substitute (3, 1) and (4, 8) for (x 1, y 1 ) and (x 2, y 2 ). Simplify.

24. Guided Practice: Example 3, continued Calculate the slope of. The slope of is. 24 1.9.3: Proving the Midsegment of a Triangle Slope formula Substitute (–2, 5) and (4, 8) for (x 1, y 1 ) and (x 2, y 2 ). Simplify.

25. Guided Practice: Example 3, continued 4.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. 25 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. 26 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. 27 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. 28 1.9.3: Proving the Midsegment of a Triangle

29. Guided Practice: Example 3, continued 5.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. 29 1.9.3: Proving the Midsegment of a Triangle

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

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