Question

Find the coordinates of the orthocenter and circumcenter of the triangle whose vertices are (0,1), (1,2) and (2,-3).
[could you please explain with diagram]

Answer 1

Answer:

A=(0,1) , B=(1,−2), C=(2,−3)

Reciprocal of slope of AB ⟹

mrAB=0−1−2−1=13

Equation of altitude CF ⟹

3(y+3)=1(x−2) ⟹

x−3y−11=0

Reciprocal of slope of BC ⟹

mrBC=1−2−3−−2=1

Equation of altitude AD ⟹

y−1=1(x−0) ⟹

y=x+1

AD X CF ⟹

x−3(x+1)−11=0 ⟹

x−3x−3−11=0 ⟹ 2x=−14 ⟹

x=−7, y=−6

Oc=(−7,−6)

Equation of perpendicular bisector of AB ⟹

−2x(0−1)+2y(−2−1)+02+12−12−(−2)2=0

2x−6y−4=0 ⟹

x−3y−2=0

Equation of perpendicular bisector of BC ⟹

−2x(1−2)+2y(−3−−2)+12+(−2)2−22−(−3)2=0

2x−2y−8=0 ⟹

x−y−4=0 ⟹

y=x−4

Meet of both perpendicular bisectors ⟹

x−3(x−4)−2=0 ⟹ x−3x+10=0

2x=10 ⟹ x=5, y=1

Cc=(5,1)

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