1 answer

What is the orthocenter of a triangle with corners at #(6 ,2 )#, #(3 ,7 )#, and (4 ,9 )#?

Question:

What is the orthocenter of a triangle with corners at #(6 ,2 )#, #(3 ,7 )#, and (4 ,9 )#?

Answers

Coordinates of orthocenter #color(blue)(O (16/11, 63/11))#

Explanation:

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Slope of BC #= m_a = (9-7)/(4-3) = 2#

Slope of AD #= -1/m_a = -1/2#

Equation of AD is

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

#2y - 4 = -x + 6#

#2y + x = 10# Eqn (1)

Slope of CA #= m_b = (9-2) / (4-6) = -(7/2)#

Slope of BE #= - (1/m_b) = 2/7#

Equation of BE is

#y - 7 = (2/7) (x - 3)#

#7y - 49 = 2x - 6#

#7y - 2x = 43# Eqn (2)

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

#color(blue)(O (16/11, 63/11))#

Confirmation:
#Slope of AB = m_c = (7-2)/(3-6) = -(5/3)#
#Slope of AD = -1/m_c = 3/5#
Equation of CF is
#y - 9 = (3/5)(x - 4)#
#5y - 3x = 33# Eqn (3)
Solving Eqns (1), (3) we get
#color(blue)(O (16/11, 63/11))#

.

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