SOLUTION: Given triangle ABC with vertices A(6, 3), B(-5, -8), and C(-3, 6), let BP and CQ be the perpendiculars dropped from B and C to their opposite sides of triangle ABC. Find the coordi

Algebra ->  Length-and-distance -> SOLUTION: Given triangle ABC with vertices A(6, 3), B(-5, -8), and C(-3, 6), let BP and CQ be the perpendiculars dropped from B and C to their opposite sides of triangle ABC. Find the coordi      Log On


   

Question 1181232: Given triangle ABC with vertices A(6, 3), B(-5, -8), and C(-3, 6), let BP and CQ be the perpendiculars dropped from B and C to their opposite sides of triangle ABC. Find the coordinates of orthocenter H of triangle ABC.
P.S. Can you show full solution, thanks.
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
Given triangle ABC with vertices
A(6, 3),
B(-5, -8), and
C(-3, 6),

let BP and CQ be the perpendiculars dropped from B and C to their opposite sides of triangle ABC
a line through BP+ is perpendicular to a line through AC
slope of a line y=mx%2Bb through AC is
m=%286-3%29%2F%28-3-6%29=-3%2F9=-1%2F3
y=-%281%2F3%29x%2Bb.......use one point to find b
3=-%281%2F3%296%2Bb
3=-2%2Bb
b=5
a line through AC is:y=-%281%2F3%29x%2B5

then a perpendicular line BP+ will have a slope -1%2F%28-1%2F3%29=3
y=3x%2Bb........use point B
-8=3%28-5%29%2Bb
-8=-15%2Bb
15-8=b
b=7
and a perpendicular line BP+ is y=3x%2B7

now we need a line that passes through CQ which is perpendicular to BA
line BA have a slope m=%28-8-3%29%2F%28-5-6%29=-11%2F-11=1
y=1x%2Bb......use one point to find b
3=1%2A6%2Bb
3-6=b
b=-3
a line that passes through BA y=x-3
a line that passes through CQ will have a slope -1%2F1=-1
y=-x%2Bb....use coordinates of C
6=-%28-3%29%2Bb
6=3%2Bb
b=3
a line that passes through CQ is y=-x%2B3


the orthocenter H is intersection point of a lines that passes through BP+ and CQ
y=3x%2B7
y=-x%2B3
--------------------
3x%2B7=-x%2B3
3x%2Bx=3-7
4x=-4
x=-1
y=3%28-1%29%2B7
y=-3%2B7
y=4
the coordinates of orthocenter H of triangle ABC are: (-1,4)