What is the orthocenter of a triangle with corners at (2,3), (5,1), and (9 ,6 )#?

1 Answer
Douglas K.
Oct 17, 2016

The Orthocenter is (12123,923)

Explanation:

Find the equation of the line that goes through the point (2,3) and is perpendicular to the line through the other two points:

y−3=9−51−6(x−2)

y−3=4−5(x−2)

y−3=−45x+85

y=−45x+235

Find the equation of the line that goes through the point (9,6) and is perpendicular to the line through the other two points:

y−6=5−23−1(x−9)

y−6=32(x−9)

y−6=32x−272

y=32x−152

The orthocenter is at the intersection of these two lines:

y=−45x+235
y=32x−152

Because y = y, we set the right sides equal and solve for the x coordinate:

32x−152=−45x+235

Multiply by 2:

3x−15=−85x+465

Multiply by 5

15x−75=−8x+46

23x=+121

#x = 121/23

y=32(12123)−152

y=32(12123)−152

y=36346−34546

y=923

The Orthocenter is (12123,923)

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