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

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Aug 3, 2018

Answer:

#color(maroon)("ortho-centre coordinates " O (73/13, 82/13)#

Explanation:

#A (9,3), B(6,9), C(2,4)#

https://www.quora.com/What-is-the-orthocentre-of-a-triangle-when-the-vertices-are-x1-y1-x2-y2-x3-y3

Slope of #bar (AB) = m_(AB) = (y_B - y_A) / (x_B - x_A) = (9-3)/ (6-9) = -2#

Slope of #bar(CF) = m_(CF) = - 1/ m(AB) = -1 / -2 = 1/2#

Equation of #bar(CF) # is #y - 4 = 1/2 (x - 2)#

#2y - x = 7# Eqn (1)

Slope of #bar (AC) = m_(AC) = (y_C - y_A) / (x_C - x_A) = (4-3)/ (2-9) = -1/7#

Slope of #bar(BE) = m_(BE) = - 1/ m(AC) = -1 / (-1/7) = 7#

Equation of #bar(BE) # is #y - 9 = 7 (x - 6)#

#7x - y = 33# Eqn (2)

Solving Eqns (1) and (2), we get the ortho-centre coordinates #O(x,y)#

#cancel(2y) - x + 14x - cancel(2y) = 7 + 66#

#x = 73/13#

#y = 164/26 = 82/13#

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