What is the orthocenter of a triangle with corners at #(3 ,1 )#, #(4 ,5 )#, and (2 ,7 )#?

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Answer 1 · 2018-01-05T12:02:39

Orthocenter coordinates #color(red)((-1/3), (-7/3)#

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Slope of #BC = m_(bc) = (y_b - y_c) / (x_b - x_c) = (5-7)/(4-2) = -1#

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

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

#color(red)(-x + y = -2)# Eqn (1)

Slope of #AB = m_(AB) = (y_a - y_b) / (x_a - x_b) = (1-5)/(3-4) = 4#

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

Equation of CF is
#y - 7 = (4) * (x - 2)#

#color(red)(-4x + y = -1)# Eqn (2)

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

Solving the two equations,
#x = -(1/3), y = -(7/3)#

Coordinates of orthocenter #color(purple)(O (-1/3, -7/3))#