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

1 Answer
Gaya3
Sep 28, 2017

(-6.bar(3),-1.bar(3))

Explanation:

Let A = (3,1)
Let B = (1,6)
Let C = (2, 2)

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(2-1)+y(2-6)=(3)(2-1)+(1)(2-6)
=>x-4y=3-4
=>color(red)(x-4y+1=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(3-2)+y(1-2)=(1)(3-2)+(6)(1-2)
=>x-y=1-6
=>color(blue)(x-y+5=0-----(2)

Equating (1) & (2):
color(red)(x-y+5)=color(blue)(x-4y+1
=>-y+4=1-5
=>color(orange)(y=-4/3-----(3)

Plugging (3) in (2):
color(blue)(x-4)color(orange)((-4/3))color(blue)(+1)=0
=>color(violet)(x=-19/3

The orthocenter is at (-19/3,-4/3) OR (-6.333...,-1.333...)
which is actually outside the triangle because the triangle is an obtuse triangle. Click here to find more.

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