Midsegments of Triangles

1. Midsegments of Triangles

2. GSP Activity Theorem 5-1 • Draw triangle ABC. • Find and construct the midpoints of segments AB and AC and label them M and N respectively. • Measure <B, <C, <AMN, and <ANM. What do you notice? • What does this tell you about segments MN and BC and how do you know this? • Measure the length of segments MN and BC and compare. Calculate BC/MN to make a comparison. • Change the size of the triangle. Does the ratio BC/MN change or stays the same?

3. 5 10 Theorem 5-1 • A segment that joins the midpoints of two sides of a triangle • is parallel to the third side. • is half as long as the third side.

4. Example problem Points D, E, and F are midpoints of the sides of the triangle shown below. What are the lengths of the sides of the triangle? DF=30, AC=50, and BC=40

5. Construct a line segment. Label the endpoints A and B. • Fold the line segment so that the endpoints lie on top of one another. Crease the fold. Mark the point where the crease intersects the line as point C and a point on the crease but not on segment AB as point D. • What do you notice about the lengths of segments AC and CB? (measure the segments if needed) • What do you notice about <ACD and <BCD? • What can you say about line CD with respect to segment AB? • Find the distance from point A to point D. Find the distance from point B to point D. What do you notice about these distances? • What can you say about the two triangles that were formed? Patty Paper Activity Theorem 5-2

6. If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment. Theorem 5-2

7. If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment. Theorem 5-3(converse of Thm 5-2)

8. Using the Perpendicular Bisector Theorem (Problem 1 on p.293)

9. Homework p. 288-289 #1-25 odd, 31, 33, 38, and 40

10. B R S A C T Review Complete the table for the figure below. Assume points R, S, and T are midpoints of the respective sides.

11. B R S A C T Review Answers

12. The distance from a point to a line (or plane) is defined to be the length of the perpendicular segment from the point to the line Distance

13. Draw an angle. The angle can be obtuse, acute, or right, but make sure the sides are fairly long. • Make a fold through the vertex of the angle so that the two sides are on top of one another. Crease the fold. Draw a line along the crease. What is this new line called with respect to the angle? • Mark point A somewhere on the angle bisector • Measure the distance from point A to each side of the angle. Recall how distance is measured from a point to a line. You may need to make folds to ensure that the line is perpendicular to the sides of the triangle. • What do you notice about these distances? Form a conjecture about a point on the angle bisector and the distance to each side of the angle. Patty Paper Activity

14. Open GSP file entitled “THM 5-4” (see wiki) • Move point P around and observe the relationship between EP and PF. Does your conjecture hold true for all locations of point P? If not, revise your conjecture. • What do you notice about the two triangles that are created? • What theorem or postulate confirms this? (SSS, ASA, SAS, AAS, or HL) Activity (continued)

15. Given: • Prove: • StatementsReasons Proof

16. If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. Theorem 5-4

17. If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle. Theorem 5-5(converse of Thm 5-4)

18. 5-2 Practice worksheet #1-19 all, 21-29 odd Homework

19. Perpendicular bisectors of a triangle GSP Activity

20. Angle bisectors of a triangle GSP Activity

21. 5-3 Triangle Bisectors

22. Concurrent • Three or more lines intersect at one point • Point of concurrency • The point at which three or more lines intersect • Circumscribed about • Inscribed in Terms

23. Perpendicular Bisector- a line, ray , or segment that is perpendicular to a segment at its midpoint Definitions of Terms

24. Circumcenter- point of concurrency of the perpendicular bisectors of a triangle • The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. • The circumcenter is the point that is the center of the circle that contains each vertex of the triangle (circle is circumscribed about the triangle) Theorem 5-6- Concurrency of Perpendicular Bisectors

25. Incenter- point of concurrency of angle bisectors in a triangle • The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle. • The incenter is the center of the circle that is inscribed in the triangle Theorem 5-7- Concurrency of Angle Bisectors

26. Medians in a Triangle GSP Activity

27. Altitudes in a Triangle GSP Activity

28. 5-4 Medians and Altitudes

30. What do these numbers represent?

31. What do these numbers represent?

36. Altitude- segment from a vertex that is perpendicular to the line that contains the opposite side • “altitude”- height or elevation (used in aviation, surveying, etc.); height above sea level of a location Definitions of Terms

37. Median- segment from a vertex to the midpoint of the opposite side • “median”- middle; divides in half Definitions of Terms

38. Centroid- point of concurrency of medians in a triangle • The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side. Theorem 5-8- Concurrency of Medians

39. Orthocenter- point of concurrency of altitudes in a triangle • The lines that contain the altitudes of a triangle are concurrent. Theorem 5-9 Concurrency of Altitudes

40. Summary

41. 5-3 Practice (12-23 all) • 5-4 Practice (1-11 all) Homework

42. p. 312-313 #8-13, 24-27 • p.304-305 #2,15-19,26-28 • p.296 #1-4, 6-8, 12-15, 18-22 • Proofs p.298 #32 and 34 Class work