5.1 Perpendiculars and Bisectors

1. 5.1 Perpendiculars and Bisectors Day 1 Part 1 CA Standard 16.0

2. Warmup • Simplify. • 1. 6x + 11y – 4x + y • 2. -5m + 3q + 4m – q • 3. -3q – 4t – 5t – 2p • 4. 9x – 22y + 18x – 3y • 5. 5x2 + 2xy – 7x2 + xy

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. • If CP is the perpendicular bisector of AB, then CA = CB. C B P A

4. Converse of the Perpendicular Bisector Theorem

5. In the diagram shown, MN is the perpendicular bisector of ST. • What segment lengths in the diagram are equal? • Explain why Q is on MN T 12 Q N M 12 S

6. Angle Bisector Theorem • 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. B D A C

7. Converse of the Angle Bisector Theorem

8. Use the diagram to answer the following. In the diagram, F is on the bisector of < DAE. • If m<BAF = 50, then m<CAF = ____ • If FC = 10, then FB = ____ • Is triangle ABF congruent to triangle ACF? Explain. A B C E D F G

10. 5.2 Bisectors of a Triangle Day 1 Part 2 CA Standards 16.0, 21.0

11. In the figure, YW bisects <XYZ. m<XYZ = 6x + 2, m<ZYW = 8x – 6. Solve for x and find m<XYZ. W Z X Y

12. 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. PA = PB = PC B P C A P is also called the circumcenter of the triangle.

13. Use the diagram shown. • E is the circumcenter of Δ ABC. • DA = ___ • BF = ___ • <EFC = ___ A E D C B F

14. Definitions • Concurrent lines: three or more lines intersect in the same point. • Point of concurrency: the point of intersection of the lines.

15. Concurrency of Angle Bisectors of a Triangle • The angle bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. PD = PE = PF B D F P C A E

16. The point of concurrency can be inside the triangle, on the triangle, or outside of the triangle. • Acute Triangle: inside • Right Triangle: on • Obtuse Triangle: outside

17. Example M • Which segments are congruent? Q R S P N L

18. Use the diagram shown. • E is the circumcenter of Δ ABC. • DA = ___ • BF = ___ • <EFC = ___ A E D C B F

19. Mini quiz on definitions… • The _____________ of the angle bisectors is called the incenter of the triangle. • If three or more lines intersect at the same point, the lines are ________. • The point of concurrency of the perpendicular bisectors of a triangle is called ____________________. Point of concurrency Concurrent Circumcenter of the triangle

20. Construction • Pg. 268 # 14, 15 • Pg. 275 # 5 – 9

21. Pg. 269 # 21 – 29, 32 • Pg. 275 # 10 - 22

22. 5.3 Medians and Altitudes of a Triangle Day 2 Part 1 CA Standards 16.0

23. Warmup • Find BD. C D 15 B A 12 12

24. 20 • AC = ___ • m<DCB = ___ • m<B = ___ 55 35 A D L B 55 20 C

25. Median of a triangle. • Median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. Median

26. The three medians of a triangle are concurrent. The point of concurrency is called the centroid of the triangle. Centroid P •

27. Concurrency of Medians of a Triangle The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. AP = 2/3 AD BP = 2/3 BF CP = 2/3 CE C F D P A B E

28. P is the centroid of ∆QRS shown. Find RT and RP when PT = 5. R P Q S T

29. Sketch ∆JKL with J(7,10), K(5,2), L(3,6). • Find the coordinates of the centroid of ∆JKL.

30. Altitude of a triangle • An altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side. Altitude

31. Every triangle has three altitudes. • The lines containing the altitudes are concurrent and intersect at a point: orthocenter of the triangle. • Where is the orthocenter located in each type of triangle? • Acute triangle • Right triangle • Obtuse triangle

32. Use the diagram shown and the given information to decide in each case whether EG is a perpendicular bisector, an angle bisector, a median, or an altitude of Δ DEF. • DG = FG • EG DF • m<DEG = m<FEG E Median T Altitude Angle bisector D F G

33. The angle bisectors of Δ ABC meet at point D. Find DE. 19 B F E L L D 19 28 C A G

35. 5.4 Midsegment Theorem Day 2 Part 2 CA Standards 17.0

36. Review Given PQ = 14, SU = 6, and QU = 3, find the perimeter of Δ STU. Q S U P R T

37. Midsegment Theorem • The segment connecting the midpoint of two sides of a triangle is parallel to the third side and is half as long. DE ll AB and DE = ½ AB C D E > > B A

39. UW and VW are midsegment of Δ RST. Find UW and RT. R 16 U T V 12 8 6 W S

40. GH, HJ and JG are midsegments of Δ EDF. • JH ll ___ • EF = ___ • DF = ___ • ___ ll DE • GH = ___ • JH = ___ 24 DF J D E 21.2 8 10.6 16 H G GH 12 8 F

42. Given the midpoints of a triangle are (7,4), (5,6) and (8,7), find the coordinates of the vertices.

43. Pg. 282 # 3 – 12 • Pg. 283 # 17 – 20 • Pg. 290 # 3 – 22, 26 – 29

44. 5.5 Inequalities in One Triangle Day 3 Part 1 CA Standards 6.0, 13.0

45. Warmup • Solve the inequality. • 1. -x + 2 > 7 • 2. c – 18 < 10 • 3. -5 + m < 21 • x – 5 > 4 • z + 6 > -2

46. Theorems • If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. m<B > m<C A 3 7 B C

47. List the angles in order from greatest to least. A 27 18 B C 23

48. If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. D 60° EF > DF E 40° F

49. Write the measurements of the triangles in order from least to greatest. J Q 7 100° R 5 45° G 6 35° H P JG, HJ, HG m<R, m<Q, m<P

50. List the sides in order from longest to shortest. F 65° 45° G E