What is the orthocenter of a triangle with corners at #(4 ,1 )#, #(6 ,2 )#, and (3 ,6 )#?

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Answer 1 · 2018-01-14T08:31:50

Coordinates of Orthocenter #color(blue)(O (56/11, 20/11))#

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Orthocenter is the concurring point of the three altitudes of a triangle and represented by ‘O’

Slope of BC #= m_a = (6-2) / (3-6) = -(4/3)#

#Slope of AD = -(1/m_a) = (3/4)#

Equation of AD is

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

#4y - 3x = -8# Eqn (1)

Slope of AB #= m_c = (2 - 1) / 6-4) = (1/2)#

Slope of CF = -(1/m_c) = -2#

Equation of CF is

#y - 6 = -2 (x - 3)#

#y + 2x = 12# Eqn (2)

Solving Eqns (1), (2)

#x = 56/11 , y = 20/11#

we get the coordinates of Orthocenter #color(blue)(O (56/11, 20/11))#

Verification

Slope #m_b = (6-1) / (3-4) = -5#

Slope of BE = -(1/m_c) = 1/5#

Equation of altitude BE is

#y - 2 = (1/5) (x - 6)#

#5y - 10 = x - 6#

#5y - x = 4# Eqn (3)

Solving equations (2) , (3),

Coordinates of #color(blue)(O (56/11, 20/11)#