SOLUTION: Determine the coordinates of the orthocentre of triangle DEF with vertices at D (-3,4) E (8,-2) F (4,5)

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Question 1137650: Determine the coordinates of the orthocentre of triangle DEF with vertices at D (-3,4) E (8,-2) F (4,5)
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

given: DEF with vertices at
D (-3,4)
E (8,-2)
F (4,5)

the orthocenter: the point where the three "altitudes" of a triangle meet
find the slopes of the lines that contain segments DE and+EF
slope of DE+=+%28-2-4%29%29%2F%288-%28-3%29%29+=+-6%2F11+
now, we need a slope of perpendicular line, altitude to segments DE:
slope of perpendicular line, altitude : -1%2Fm+=+-1%2F%28-6%2F11%29 -> m+=+11%2F6
y+=+mx+%2B+b (substitute m+=+11%2F6, x+=+4, y+=+5)
5+=+%2811%2F6%29%284%29+%2B+b
5+=+44%2F6+%2B+b
b+=+5-22%2F3
b+=+15%2F3-22%2F3
b=+-7%2F3

equation of the altitude to DE:
+y+=++%2811%2F6%29x+-7%2F3

slope of EF+=+%28-2-4%29%2F%288-5%29+=+-7%2F4+
altitude to EF : +-1%2Fm+=+-1%2F%28-7%2F4%29 -> m+=+4%2F7
y+=+mx+%2B+b (substitute m+=+4%2F7, x+=+-3, y+=4)
4=+-3%284%2F7+%29+%2B+b
4+=+-12%2F7+%2B+b
b+=+4%2B12%2F7
b=+40%2F7
equation of the altitude to EF:
y+=+%284%2F7%29+x+%2B40%2F7

intersection of the altitudes will be orthocenter
equation 1: y+=++%2811%2F6%29x+-7%2F3
equation 2: y+=%284%2F7%29+x+%2B40%2F7
Solving for x and y:
%2811%2F6%29x+-7%2F3=%284%2F7%29+x+%2B40%2F7
%2811%2F6%29x+-%284%2F7%29+x+=+7%2F3%2B40%2F7
%2853+x%29%2F42=+169%2F21
x+=+338%2F53
x=+6.4
y+=++%2811%2F6%29x+-7%2F3
y+=++%2811%2F6%29%286.4%29+-7%2F3
y=+9.4
The coordinates are (6.4, 9.4). This is the orthocenter.