15. The vertices of A DEF are D(0,0), E(0,7), and F(6,3).
Find the coordinates of the orthocenter of A DEF.

Orthocenter of a triangle with one vertex at the origin D(0,0), E(0,7), F(6,3)

The orthocenter is the meet of the altitudes.  

DE is the line x=0.  That's the y axis.  The altitude is the perpendicular through F(6,3), so is the horizontal line y=3.  

DF is the line

6y=3x

y=(1/2)x

Perpendicular through E(0,7) is

y - 7 = (-2)(x - 0)

y = -2x + 7

That's the other altitude.  Intersecting with y=3

3 = -2x + 7

-4 = -2x

x = 2

y = 3

orthocenter is (2,3)

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