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

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Answer 1 · 2018-01-16T08:21:21

Coordinates of orthocenter ‘O’ is #color(red)(1,2)#

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Slope of AB #m_(AB) = (3-7) / (2 - 5) = 4/3#

Slope of CF = m_(CF) = -(1/m_(AB)) = -(1/ (4/3)) = -(3/4)#

Equation of CF is

#y - 2 = -(3/4) * (x - 1)#

#4y - 8 = -3x + 3#

#4y + 3x = 11# Eqn (1)

Slope of BC #= m_(BC) = (2-3) / (1-2) = 1#

Slope of AD #= m_(AD) = -(1/ m_(BC) )= -1#

Equation of AD is

#y - 2 = -1 * (x - 1)#

#y + x = 3# Eqn (2)

Solving Eqns (1), (2) will get us the coordinates of orthocenter ‘O’

#x = 1, y = 2#

Coordinates of orthocenter ‘O’ is #color(red)(1,2)#