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Triangle Centres

Triangle Centres

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Triangle Centres

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  1. Triangle Centres

  2. Mental Health Break

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

  4. Centroid • Equation of AD (median) • Strategy…. • Find midpoint D • 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. Centroid Equation of AD (median)

  6. Centroid • Equation of BE (median) • Strategy…. • Find midpoint E • 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. Centroid Equation of BE (median)

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

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

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

  11. Centroid Equation of median AD Equation of median BE

  12. Centroid • Equation of AD (median) • Strategy…. • Find midpoint D • 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. Centroid – Intersection of Eq’n AD & BE AD BE 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. Orthocentre • Equation of altitude AD • Strategy…. • Find “m” of BC • Take –ve reciprocal of “m” of BC to get “m” of AD • 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. Orthocentre Equation of altitude AD

  27. Orthocentre • Equation of altitude BE • Strategy…. • Find “m” of AC • Take –ve reciprocal of “m” of AC to get “m” of BE • 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. Orthocentre Equation of altitude BE

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

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

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

  32. Orthocentre 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. Orthocentre – Intersection of Eq’n AD & BE AD BE 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. Circumcenter • Equation of ED (perpendicular bisector) • Strategy… (use A (-1, 4), B (-1, -2) & C(5, 1)) • Find midpoint D • 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. Circumcenter Equation of ED (perpendicular bisector)

  44. Circumcenter • Equation of FG (perpendicular bisector) • Strategy… (use A (-1, 4), B (-1, -2) & C(5, 1)) • Find midpoint F • 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. Circumcenter Question? Do we have to find the equation of perpendicular bisector HI?

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

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

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

  49. Circumcenter Equation of FG (perpendicular bisector)

  50. Circumcentre eq’n ED – Midpoint of BC