Find the equation of the line that goes through the point (2,3)(2,3) and is perpendicular to the line through the other two points:
y - 3 = (9 - 5)/(1 -6)(x - 2)y−3=9−51−6(x−2)
y - 3 = (4)/(-5)(x - 2)y−3=4−5(x−2)
y - 3 = -4/5x + 8/5y−3=−45x+85
y = -4/5x + 23/5y=−45x+235
Find the equation of the line that goes through the point (9,6)(9,6) and is perpendicular to the line through the other two points:
y - 6 = (5 - 2)/(3 - 1)(x - 9)y−6=5−23−1(x−9)
y - 6 = (3)/(2)(x - 9)y−6=32(x−9)
y - 6 = 3/2x - 27/2y−6=32x−272
y = 3/2x - 15/2y=32x−152
The orthocenter is at the intersection of these two lines:
y = -4/5x + 23/5y=−45x+235
y = 3/2x - 15/2y=32x−152
Because y = y, we set the right sides equal and solve for the x coordinate:
3/2x - 15/2 = -4/5x + 23/532x−152=−45x+235
Multiply by 2:
3x - 15 = -8/5x + 46/53x−15=−85x+465
Multiply by 5
15x - 75 = -8x + 4615x−75=−8x+46
23x = + 12123x=+121
#x = 121/23
y = 3/2(121/23) - 15/2y=32(12123)−152
y = 3/2(121/23) - 15/2y=32(12123)−152
y = 363/46 - 345/46y=36346−34546
y = 9/23y=923
The Orthocenter is (121/23, 9/23)(12123,923)
