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Properties of Triangles

Chapter 5. Properties of Triangles. Vocab Review. Intersect Midpoint Angle Bisector Perpendicular Bisector Construction of a Perpendicular through a point on a line. Perpendicular Bisector Theorem. If a point is on the perpendicular bisector of a segment,

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Properties of Triangles

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  1. Chapter 5 Properties of Triangles

  2. Vocab Review • Intersect • Midpoint • Angle Bisector • Perpendicular Bisector • Construction of a Perpendicular through a point on a line

  3. Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment

  4. Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of the segment , then it is on the perpendicular bisector of a segment

  5. Using Perpendicular Bisectors • What segment lengths are equal? • Why is L on OQ? Given: OQ is the  bisector of MP

  6. Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the two side of the angle

  7. Converse of the Angle Bisector Theorem 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

  8. Angle Bisector Theorem Let’s Prove it… Given: • D is on the bisector of  BAC • BD  AB • CD  AC Prove: DB  DC

  9. CHECK POINT “Guided Practice” Page 267 #1-7

  10. Perpendicular Bisector of a Triangle Line (or ray/ segments) that is perpendicular to a side of the triangle at the midpoint of the side. Concurrent @ Circumcenter (not necessarily inside the triangle)

  11. Perpendicular Bisector of a Triangle

  12. Concurrency • Point of Concurrency • The point of intersection of the lines • Concurrent Lines • When 3 or more lines (rays/segments) intersect at the same point. .

  13. Concurrency of Perpendicular Bisectors of a Triangle The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle Circumcenter AG = BG = CG

  14. Exit Ticket

  15. Angle Bisector of a Triangle Bisector of an angle of the triangle. (one for each angle) Concurrent @ Incenter (always inside the triangle)

  16. Angle Bisector of a Triangle

  17. Concurrency of Angle Bisectors of a Triangle The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle Incenter EG = DG = FG

  18. Pythagorean Theorem a2 + b2 = c2

  19. Pythagorean Theorem a2 + b2 = c2 EXAMPLE: a= ?, b= 15, c= 17 a2 + b2 = c2 a2 + 152 = 172 a2 + 225 = 289 a2 = 64 a = 8

  20. CHECK POINT Classwork Page 275-278 #10-23,

  21. Median of a Triangle Segment whose end points are a vertex of the triangle and the midpoint of the opposite side. Concurrent @ Centroid (always inside the triangle)

  22. Median of a Triangle

  23. Concurrency of Medians of a Triangle Centoid The medians of a triangle intersect at a point that is ²/₃ of the distance from each vertex to the midpoint of the opposite side. AG = ²/₃AD, BG = ²/₃BE, CG = ²/₃CF

  24. EXAMPLE Page 280 Example #1

  25. Altitude of a Triangle Perpendicular segment from a vertex to the opposite side. Concurrent @ Orthocenter (inside, on, or outside the triangle)

  26. EXAMPLE Page 281 Example #3

  27. Concurrency of Altitudes of a Triangle orthocenter Lines containing the altitudes of a triangle are concurrent Intersect at some point (H) called the orthocenter (Inside, On, or Outside)

  28. Video Activity

  29. CHECK POINT Classwork: pg 282 #13-16 Homework: pg 282 #1-11, 17-20, & 34

  30. Warm-up Find the coordinates of the midpoint: 1. (2, 0) and (0, 2) 2. (5, 8) and (-3, -4) Find the slope of the line through the given points: 1. (3, 11) and (3, 4) 2. (-2, -6) and (4, -9) 3. (4, 3) and (8, 3)

  31. Midsegment of a Triangle *Segment that connects the midpoints of two sides of a triangle *Half as long as the side of the triangle it is parallel to

  32. ** REVIEW ** MIDPOINT FORMULA Midpoint =

  33. Midsegment of a Triangle • Find a Midsegment of the triangle…. Let D be the midpoint of AB Let E be the midpoint of BC

  34. MidsegmentTheorem The segment connecting the midpoint of two sides of a triangle is parallel to the third side and is half as long DEBC and DE=¹⁄₂BC

  35. Prove it… How do you prove DEBC? Use the slope formula to find each slope (parallel lines have the same slope) How do you prove DE=¹⁄₂BC? Use the distance formula to find the distance of each segment

  36. Exit Ticket Pg 290 #3-9

  37. CHECK POINT Classwork: pg 290 #10-18, 21-25

  38. Warm-up In ∆DEF, G, H, and J are the midpoints of the sides… H D E 1. Which segment is parallel to EF? GH J G 2. If GJ=3, what is HE? 3 2.5 F 3. If DF=5, what is HJ?

  39. Test Review Pgs 310-311 #1-15 Pgs 313 #1-11

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