What is the orthocenter of a triangle with corners at #(3 ,6 )#, #(4 ,2 )#, and (5 ,7 )#?

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Answer 1 · 2017-12-28T06:36:35

Orthocenter of the triangle #color(purple)(O (17/9, 56/9))#

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Slope of #BC = m_(bc) = (y_b - y_c) / (x_b - x_c) = (2-7)/4-5) = 5#

Slope of #AD = m_(ad) = - (1/m_(bc) = - (1/5)#

Equation of AD is
#y - 6 = -(1/5) * (x - 3)#

#color(red)(x + 5y = 33)# Eqn (1)

Slope of #AB = m_(AB) = (y_a - y_b) / (x_a - x_b) = (6-2)/(3-4) = -4#

Slope of #CF = m_(CF) = - (1/m_(AB) = - (1/-4) = 4#

Equation of CF is
#y - 7 = (1/4) * (x - 5)#

#color(red)(-x + 4y = 23)# Eqn (2)

Solving Eqns (1) & (2), we get the orthocenter #color(purple)(O)# of the triangle

Solving the two equations,
#x = 17/9, y = 56/9#

Coordinates of orthocenter #color(purple)(O (17/9, 56/9))#