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Triangle Centres. Mental Health Break. Given the following triangle, find the: centroid orthocenter circumcenter. 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”
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Given the following triangle, find the: • centroid • orthocenter • circumcenter
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!
Centroid Equation of AD (median)
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”
Centroid Equation of BE (median)
Centroid Question? Do we have to find the equation of median CF also?
Centroid No We only need the equations of 2 medians… So, what do we do now?
Centroid We need to find the Point of Intersection for medians AD & BE using either substitution or elimination
Centroid Equation of median AD Equation of median BE
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”
Centroid – Intersection of Eq’n AD & BE AD BE Add AD and BE Simplify and solve for y
Centroid – Intersection of Eq’n AD & BE Substitute y = 1 into one of the equations Therefore, the point of intersection is (1,1)
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”
Centroid – Intersection of Eq’n AD & BE Therefore, the Centroid is (1,1)
Orthocentre Equation of altitude AD
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”
Orthocentre Equation of altitude BE
Orthocentre Question? Do we have to find the equation of altitude CF also?
Orthocentre No We only need the equations of 2 altitudes… So, what do we do now?
Orthocentre We need to find the Point of Intersection for altitudes AD & BE using either substitution or elimination
Orthocentre Equation of altitude AD Equation of altitude BE
Orthocentre – Intersection of Eq’n AD & BE AD BE Add AD and BE Simplify and solve for y
Orthocentre – Intersection of Eq’n AD & BE Substitute y = 1 into one of the equations Therefore, the point of intersection or Orthocentre
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”
Circumcenter Equation of ED (perpendicular bisector)
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”
Circumcenter Question? Do we have to find the equation of perpendicular bisector HI?
Circumcenter No We only need the equations of 2 perpendicular bisectors… So, what do we do now?
Circumcenter We need to find the Point of Intersection for perpendicular bisectors ED & FG using either substitution or elimination
Circumcenter Equation of perpendicular bisector ED Equation of perpendicular bisector FG
Circumcenter Equation of FG (perpendicular bisector)