2. Proving the Midsegment of a Triangle Adapted from Walch Education

3. Key Concepts In the diagram, the midpoint of is X. The midpoint of is Y. A midsegment of is. 21.9.3: Proving the Midsegment of a Triangle 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.

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. Every triangle has three midsegments. 31.9.3: Proving the Midsegment of a Triangle

5. 4 Theorem Triangle Midsegment Theorem A midsegment of a triangle is parallel to the third side and is half as long.

6. Key Concepts, continued When all three of the midsegments of a triangle are connected, a midsegment triangle is created. 51.9.3: Proving the Midsegment of a Triangle

7. Key Concepts, continued Coordinate proofs, proofs that involve calculations and make reference to the coordinate plane, are often used to prove many theorems. 61.9.3: Proving the Midsegment of a Triangle

8. Practice 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. 71.9.3: Proving the Midsegment of a Triangle

9. Plot the midpoints on a coordinate plane. 81.9.3: Proving the Midsegment of a Triangle

10. Connect the midpoints to form the midsegments 91.9.3: Proving the Midsegment of a Triangle

11. Calculate the slope of each midsegment  Calculate the slope of.  The slope of is. 101.9.3: 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.

12. Calculate the slope of each midsegment Calculate the slope of.  The slope of is 7. 111.9.3: 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.

13. Calculate the slope of each midsegment Calculate the slope of.  The slope of is. 121.9.3: 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.

14. 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. 131.9.3: Proving the Midsegment of a Triangle

15. Draw the lines that contain the midpoints. The slope of is. From point Y, draw a line that has a slope of. 141.9.3: Proving the Midsegment of a Triangle

16. Draw the lines that contain the midpoints. The slope of is 7. From point X, draw a line that has a slope of 7. 151.9.3: Proving the Midsegment of a Triangle

17. Draw the lines that contain the midpoints. 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. 161.9.3: Proving the Midsegment of a Triangle

18. 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. 171.9.3: Proving the Midsegment of a Triangle

19. 181.9.3: Proving the Midsegment of a Triangle

20. Thanks for Watching! Ms. Dambreville