Triangle Centres. Mental Health Break Given the following triangle, find the:  centroid  orthocenter  circumcenter.

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Presentation transcript:

Triangle Centres

Mental Health Break

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

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!

Equation of AD (median)

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”

Equation of BE (median)

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

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

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

Equation of median AD Equation of median BE

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”

Centroid Eq’n AD – Midpoint of BC

Centroid Eq’n AD – Slope of AD

Centroid Eq’n AD – Finding “b”

Centroid Eq’n AD – Equation

Centroid Eq’n BE – Midpoint of AC

Centroid Eq’n BE – Slope of BE

Centroid Eq’n BE – Finding “b”

Centroid Eq’n BE – Equation

Centroid – Intersection of Eq’n AD & BE

BE AD 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)

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”

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

Equation of altitude AD

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”

Equation of altitude BE

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

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

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

Equation of altitude AD Equation of altitude BE

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

Orthocentre Eq’n AD – Finding “b”

Orthocentre Eq’n AD – Equation

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

Orthocentre Eq’n BE – Finding “b”

Orthocentre Eq’n BE – Equation

Orthocentre – Intersection of Eq’n AD & BE

BE AD 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

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”

Equation of ED (perpendicular bisector)

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”

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

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

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

Equation of perpendicular bisector ED Equation of perpendicular bisector FG

Equation of FG (perpendicular bisector)

Circumcentre eq’n ED – Midpoint of BC

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

Circumcentre Eq’n ED – Finding “b”

Circumcentre Eq’n ED – Equation

Circumcentre Eq’n FG – Midpoint of AC

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

Circumcentre Eq’n FG – Finding “b”

Circumcentre Eq’n FG – Equation

Circumcentre – Intersection of Eq’n ED & FG

FG ED Add ED and FG 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 Circumcentre