What is the orthocenter of a triangle with corners at #(5 ,2 )#, #(3 ,7 )#, and (0 ,9 )#?

1 Answer
sankarankalyanam
Oct 24, 2017

Coordinates of orthocenter #(9/11, -47/11)#

Explanation:

#Let# #A = (5,2)#
#Let# #B = (3,7)#
#Let# #C = (0,9)#

Equation for altitude through A:
#x(x_3-x_2)+y(y_3-y_2)=x_1(x_3-x_2)+y1(y_3-y_2)#
#=>x(0-3)+y(9-7)=(5)(0-3)+(2)(9-7)#
#=>-3x + 2y = -15 + 4#
#=>color(red)(3x - 2y + 11 = 0)#-----(1)

Equation for altitude through B:
#x(x_1-x_3)+y(y_1-y_3)=x_2(x_1-x_3)+y2(y_1-y_3)#
#=>x(5-0)+y(2-9)=(3)(5-0)+(7)(2-9)#
#=>5x -7y=15-49#
#=>color(blue)(5x - 7y -34 = 0#-----(2)

Equating (1) & (2):
#color(red)(3x - 2y +1 1 =color(blue)(5x - 7y -34)#
#=>color(orange)(y=-47/11)#-----(3)

Plugging (3) in (2):
#=>color(violet)(x= 9/11#

The orthocenter is at #(9/11, -47/11)#
which is actually outside the #triangle# because the #triangle# is an obtuse one#

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