5.1 midsegments of triangles

1. 5.1 midsegments of triangles Geometry Mrs. Spitz Fall 2004

2. Objectives: • Use properties of midsegments to solve problems • Use properties of perpendicular bisectors • Use properties of angle bisectors to identify equal distances such as the lengths of beams in a room truss.

3. Midsegments of triangles • Turn in your book to page 243. In the blue box is an investigation. • Complete the investigation, we’ll share our conjectures in 3 minutes.

4. Midsegment • A segment that connects the midpoints of two sides • Midsegment theorem: if a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side AND is half it’s length.

5. Example 1 • In triangle XYZ, M, N, and P are midpoints. The perimeter of triangle MNP is 60. Find NP and YZ. X 22 P M 24 Y N Z

6. Example 2 • In triangle DEF, A, B, and C are midpoints. Name pairs of parallel segments.

7. X U = = Y | | Z • Check for understanding. • AB= 10 and CD = 18. Find EB, BC, and AC. • Critical Thinking • Find m<VUZ. Justify your answer A = | E B | = D C V

8. Real world connection • CD is a new bridge being built over a lake as shown. Find the length of the bridge.

9. Assignment: • Page 246 #1-20 we’ll go over them then you’ll turn in tomorrow #’s 21-36; 47-55

10. 5.2 Bisectors in Triangles

11. Objectives: • Use properties of midsegments to solve problems • Use properties of perpendicular bisectors • Use properties of angle bisectors to identify equal distances such as the lengths of beams in a room truss.

12. In lesson 1.5, you learned that a segment bisector intersects a segment at its midpoint. A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector. Use Properties of perpendicular bisectors CP is a  bisector of AB

13. Equidistant • A point is equidistant from two points if its distance from each point is the same. In the construction above, C is equidistant from A and B because C was drawn so that CA = CB.

14. Perpendicular bisector theorem • If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

15. If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If CP is the perpendicular bisector of AB, then CA = CB. Theorem 5.1 Perpendicular Bisector Theorem

16. Converse of the perpendicular bisector theorem • If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

17. If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. If DA = DB, then D lies on the perpendicular bisector of AB. Theorem 5.2: Converse of the Perpendicular Bisector Theorem

18. Refer to the diagram for Theorem 5.1. Suppose that you are given that CP is the perpendicular bisector of AB. Show that right triangles ∆ABC and ∆BPC are congruent using the SAS Congruence Postulate. Then show that CA ≅ CB. Plan for Proof of Theorem 5.1

19. Statements: CP is perpendicular bisector of AB. CP  AB AP ≅ BP CP ≅ CP CPB ≅ CPA ∆APC ≅ ∆BPC CA ≅ CB Reasons: Given Given: CP is perpendicular to AB. Prove: CA≅CB

20. Statements: CP is perpendicular bisector of AB. CP  AB AP ≅ BP CP ≅ CP CPB ≅ CPA ∆APC ≅ ∆BPC CA ≅ CB Reasons: Given Definition of Perpendicular bisector Given: CP is perpendicular to AB. Prove: CA≅CB

21. Statements: CP is perpendicular bisector of AB. CP  AB AP ≅ BP CP ≅ CP CPB ≅ CPA ∆APC ≅ ∆BPC CA ≅ CB Reasons: Given Definition of Perpendicular bisector Given Given: CP is perpendicular to AB. Prove: CA≅CB

22. Statements: CP is perpendicular bisector of AB. CP  AB AP ≅ BP CP ≅ CP CPB ≅ CPA ∆APC ≅ ∆BPC CA ≅ CB Reasons: Given Definition of Perpendicular bisector Given Reflexive Prop. Congruence. Given: CP is perpendicular to AB. Prove: CA≅CB

23. Statements: CP is perpendicular bisector of AB. CP  AB AP ≅ BP CP ≅ CP CPB ≅ CPA ∆APC ≅ ∆BPC CA ≅ CB Reasons: Given Definition of Perpendicular bisector Given Reflexive Prop. Congruence. Definition right angle Given: CP is perpendicular to AB. Prove: CA≅CB

24. Statements: CP is perpendicular bisector of AB. CP  AB AP ≅ BP CP ≅ CP CPB ≅ CPA ∆APC ≅ ∆BPC CA ≅ CB Reasons: Given Definition of Perpendicular bisector Given Reflexive Prop. Congruence. Definition right angle SAS Congruence Given: CP is perpendicular to AB. Prove: CA≅CB

25. Statements: CP is perpendicular bisector of AB. CP  AB AP ≅ BP CP ≅ CP CPB ≅ CPA ∆APC ≅ ∆BPC CA ≅ CB Reasons: Given Definition of Perpendicular bisector Given Reflexive Prop. Congruence. Definition right angle SAS Congruence CPCTC Given: CP is perpendicular to AB. Prove: CA≅CB

26. In the diagram MN is the perpendicular bisector of ST. What segment lengths in the diagram are equal? Explain why Q is on MN. Ex. 1 Using Perpendicular Bisectors

27. What segment lengths in the diagram are equal? Solution: MN bisects ST, so NS = NT. Because M is on the perpendicular bisector of ST, MS = MT. (By Theorem 5.1). The diagram shows that QS = QT = 12. Ex. 1 Using Perpendicular Bisectors

28. Explain why Q is on MN. Solution: QS = QT, so Q is equidistant from S and T. By Theorem 5.2, Q is on the perpendicular bisector of ST, which is MN. Ex. 1 Using Perpendicular Bisectors

29. If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. If mBAD = mCAD, then DB = DC Angle Bisector Theorem

30. If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle. If DB = DC, then mBAD = mCAD. Converse of the Angle Bisector Theorem

31. Given: D is on the bisector of BAC. DB AB, DC  AC. Prove: DB = DC Plan for Proof: Prove that ∆ADB ≅ ∆ADC. Then conclude that DB ≅DC, so DB = DC. Ex. 2: Proof of angle bisector theorem

32. By definition of an angle bisector, BAD ≅ CAD. Because ABD and ACD are right angles, ABD ≅ ACD. By the Reflexive Property of Congruence, AD ≅ AD. Then ∆ADB ≅ ∆ADC by the AAS Congruence Theorem. By CPCTC, DB ≅ DC. By the definition of congruent segments DB = DC. Paragraph Proof

33. Given: D is in the interior of ABC and is equidistant from BA and BC. Prove: D lies on the angle bisector of ABC. Developing Proof

34. Statements: D is in the interior of ABC. D is ___?_ from BA and BC. ____ = ____ DA  ____, ____  BC __________ __________ BD ≅ BD __________ ABD ≅ CBD BD bisects ABC and point D is on the bisector of ABC Given: D is in the interior of ABC and is equidistant from BA and BC. Prove: D lies on the angle bisector of ABC. Reasons: • Given

35. Statements: D is in the interior of ABC. D is EQUIDISTANT from BA and BC. ____ = ____ DA  ____, ____  BC __________ __________ BD ≅ BD __________ ABD ≅ CBD BD bisects ABC and point D is on the bisector of ABC Given: D is in the interior of ABC and is equidistant from BA and BC. Prove: D lies on the angle bisector of ABC. Reasons: • Given • Given

36. Statements: D is in the interior of ABC. D is EQUIDISTANT from BA and BC. DA = DC DA  ____, ____  BC __________ __________ BD ≅ BD __________ ABD ≅ CBD BD bisects ABC and point D is on the bisector of ABC Given: D is in the interior of ABC and is equidistant from BA and BC. Prove: D lies on the angle bisector of ABC. Reasons: • Given • Given • Def. Equidistant

37. Statements: D is in the interior of ABC. D is EQUIDISTANT from BA and BC. DA = DC DA  _BA_, __DC_  BC __________ __________ BD ≅ BD __________ ABD ≅ CBD BD bisects ABC and point D is on the bisector of ABC Given: D is in the interior of ABC and is equidistant from BA and BC. Prove: D lies on the angle bisector of ABC. Reasons: • Given • Given • Def. Equidistant • Def. Distance from point to line.

38. Statements: D is in the interior of ABC. D is EQUIDISTANT from BA and BC. DA = DC DA  _BA_, __DC_  BC DAB = 90°DCB = 90° __________ BD ≅ BD __________ ABD ≅ CBD BD bisects ABC and point D is on the bisector of ABC Given: D is in the interior of ABC and is equidistant from BA and BC. Prove: D lies on the angle bisector of ABC. Reasons: • Given • Given • Def. Equidistant • Def. Distance from point to line. • If 2 lines are , then they form 4 rt. s.

39. Statements: D is in the interior of ABC. D is EQUIDISTANT from BA and BC. DA = DC DA  _BA_, __DC_  BC DAB and DCB are rt. s DAB = 90°DCB = 90° BD ≅ BD __________ ABD ≅ CBD BD bisects ABC and point D is on the bisector of ABC Given: D is in the interior of ABC and is equidistant from BA and BC. Prove: D lies on the angle bisector of ABC. Reasons: • Given • Given • Def. Equidistant • Def. Distance from point to line. • If 2 lines are , then they form 4 rt. s. • Def. of a Right Angle

40. Statements: D is in the interior of ABC. D is EQUIDISTANT from BA and BC. DA = DC DA  _BA_, __DC_  BC DAB and DCB are rt. s DAB = 90°DCB = 90° BD ≅ BD __________ ABD ≅ CBD BD bisects ABC and point D is on the bisector of ABC Given: D is in the interior of ABC and is equidistant from BA and BC. Prove: D lies on the angle bisector of ABC. Reasons: • Given • Given • Def. Equidistant • Def. Distance from point to line. • If 2 lines are , then they form 4 rt. s. • Def. of a Right Angle • Reflexive Property of Cong.

41. Statements: D is in the interior of ABC. D is EQUIDISTANT from BA and BC. DA = DC DA  _BA_, __DC_  BC DAB and DCB are rt. s DAB = 90°DCB = 90° BD ≅ BD ∆ABD ≅ ∆CBD ABD ≅ CBD BD bisects ABC and point D is on the bisector of ABC Given: D is in the interior of ABC and is equidistant from BA and BC. Prove: D lies on the angle bisector of ABC. Reasons: • Given • Given • Def. Equidistant • Def. Distance from point to line. • If 2 lines are , then they form 4 rt. s. • Def. of a Right Angle • Reflexive Property of Cong. • HL Congruence Thm.

42. Statements: D is in the interior of ABC. D is EQUIDISTANT from BA and BC. DA = DC DA  _BA_, __DC_  BC DAB and DCB are rt. s DAB = 90°DCB = 90° BD ≅ BD ∆ABD ≅ ∆CBD ABD ≅ CBD BD bisects ABC and point D is on the bisector of ABC Given: D is in the interior of ABC and is equidistant from BA and BC. Prove: D lies on the angle bisector of ABC. Reasons: • Given • Given • Def. Equidistant • Def. Distance from point to line. • If 2 lines are , then they form 4 rt. s. • Def. of a Right Angle • Reflexive Property of Cong. • HL Congruence Thm. • CPCTC

43. Statements: D is in the interior of ABC. D is EQUIDISTANT from BA and BC. DA = DC DA  _BA_, __DC_  BC DAB and DCB are rt. s DAB = 90°DCB = 90° BD ≅ BD ∆ABD ≅ ∆CBD ABD ≅ CBD BD bisects ABC and point D is on the bisector of ABC Given: D is in the interior of ABC and is equidistant from BA and BC. Prove: D lies on the angle bisector of ABC. Reasons: • Given • Given • Def. Equidistant • Def. Distance from point to line. • If 2 lines are , then they form 4 rt. s. • Def. of a Right Angle • Reflexive Property of Cong. • HL Congruence Thm. • CPCTC • Angle Bisector Thm.

44. Assignment: • You’ll do page 251 #1-11 we’ll go over them then you’ll turn #’s 12-31; 41, 42, 55-63