1. Triangle Centres

2. Mental Health Break

3. Given the following triangle, find the:  centroid  orthocenter  circumcenter

4. Equation of AD (median) Strategy…. 1.Find midpoint D 2.Find eq’n of AD by -Find slope “m” of AD using A & D -Plug “m” & point A or D into y=mx+b & solve for “b” -Now write eq’n using “m” & “b” Remember – the centroid is useful as the centre of the mass of a triangle – you can balance a triangle on a centroid!

5. Equation of AD (median)

6. Equation of BE (median) Strategy…. 1.Find midpoint E 2.Find eq’n of BE by -Find slope “m” of BE using B & E -Plug “m” & point B or E into y=mx+b & solve for “b” -Now write eq’n using “m” & “b”

7. Equation of BE (median)

8. Question? Do we have to find the equation of median CF also?

9. No We only need the equations of 2 medians… So, what do we do now?

10. We need to find the Point of Intersection for medians AD & BE using either substitution or elimination

11. Equation of median AD Equation of median BE

12. Equation of AD (median) Strategy…. 1.Find midpoint D 2.Find eq’n of AD by -Find slope “m” of AD using A & D -Plug “m” & point A or D into y=mx+b & solve for “b” -Now write eq’n using “m” & “b”

13. Centroid Eq’n AD – Midpoint of BC

14. Centroid Eq’n AD – Slope of AD

15. Centroid Eq’n AD – Finding “b”

16. Centroid Eq’n AD – Equation

17. Centroid Eq’n BE – Midpoint of AC

18. Centroid Eq’n BE – Slope of BE

19. Centroid Eq’n BE – Finding “b”

20. Centroid Eq’n BE – Equation

21. Centroid – Intersection of Eq’n AD & BE

22. BE AD Add AD and BE Simplify and solve for y

23. Centroid – Intersection of Eq’n AD & BE Substitute y = 1 into one of the equations Therefore, the point of intersection is (1,1)

24. Equation of altitude AD Strategy…. 1.Find “m” of BC 2.Take –ve reciprocal of “m” of BC to get “m” of AD 3.Find eq’n of AD by -Plug “m” from 2. & point A into y=mx+b & solve for “b” -Now write eq’n using “m” & “b”

25. Centroid – Intersection of Eq’n AD & BE Therefore, the Centroid is (1,1)

26. Equation of altitude AD

27. Equation of altitude BE Strategy…. 1.Find “m” of AC 2.Take –ve reciprocal of “m” of AC to get “m” of BE 3.Find eq’n of BE by -Plug “m” from 2. & point B into y=mx+b & solve for “b” -Now write eq’n using “m” & “b”

28. Equation of altitude BE

29. Question? Do we have to find the equation of altitude CF also?

30. No We only need the equations of 2 altitudes… So, what do we do now?

31. We need to find the Point of Intersection for altitudes AD & BE using either substitution or elimination

32. Equation of altitude AD Equation of altitude BE

33. Orthocentre Eq’n AD – Slope of BC then Slope of AD

34. Orthocentre Eq’n AD – Finding “b”

35. Orthocentre Eq’n AD – Equation

36. Orthocentre Eq’n BE – Slope of AC then Slope of BE

37. Orthocentre Eq’n BE – Finding “b”

38. Orthocentre Eq’n BE – Equation

39. Orthocentre – Intersection of Eq’n AD & BE

40. BE AD Add AD and BE Simplify and solve for y

41. Orthocentre – Intersection of Eq’n AD & BE Substitute y = 1 into one of the equations Therefore, the point of intersection or Orthocentre

42. Equation of ED (perpendicular bisector) Strategy… (use A (-1, 4), B (-1, -2) & C(5, 1)) 1.Find midpoint D 2.Find eq’n of ED by -Find slope “m” of BC using B & E -Take –ve reciprocal to get “m” of ED -Plug “m” ED & point D into y = mx+b & solve for b -Now write eq’n using “m” & “b”

43. Equation of ED (perpendicular bisector)

44. Equation of FG (perpendicular bisector) Strategy… (use A (-1, 4), B (-1, -2) & C(5, 1)) 1.Find midpoint F 2.Find eq’n of ED by -Find slope “m” of AC using A & C -Take –ve reciprocal to get “m” of FG -Plug “m” FG & point F into y = mx+b & solve for b -Now write eq’n using “m” & “b”

45. Question? Do we have to find the equation of perpendicular bisector HI?

46. No We only need the equations of 2 perpendicular bisectors… So, what do we do now?

47. We need to find the Point of Intersection for perpendicular bisectors ED & FG using either substitution or elimination

48. Equation of perpendicular bisector ED Equation of perpendicular bisector FG

49. Equation of FG (perpendicular bisector)

50. Circumcentre eq’n ED – Midpoint of BC

51. Circumcentre Eq’n ED – Slope of BC & Slope of ED

52. Circumcentre Eq’n ED – Finding “b”

53. Circumcentre Eq’n ED – Equation

54. Circumcentre Eq’n FG – Midpoint of AC

55. Circumcentre Eq’n FG – Slope of AC & Slope of FG

56. Circumcentre Eq’n FG – Finding “b”

57. Circumcentre Eq’n FG – Equation

58. Circumcentre – Intersection of Eq’n ED & FG

59. FG ED Add ED and FG Simplify and solve for y

60. Orthocentre – Intersection of Eq’n AD & BE Substitute y = 1 into one of the equations Therefore, the point of intersection or Circumcentre