What is the orthocenter of a triangle with corners at (4 ,1 ), (7 ,4 ), and (2 ,8 )#?

1 Answer
Jan 22, 2018

(53/18, 71/18)

Explanation:

1) Find the slope of two lines.
(4,1) and (7,4)
m_1 = 1
(7,4) and (2,8)
m_2 = -4/5

2) Find the perpendicular of both slopes.
m_(perp1) = -1
m_(perp2) = 5/4

3) Find the midpoints of the points you used.
(4,1) and (7,4)
mid_1 = (11/2,3/2)
(7,4) and (2,8)
mid_2 = (9/2,6)

4) Using the slope, find an equation that fits it.
m=-1, point = (11/2, 3/2)
y=-x+b
3/2=-11/2+b
b=7

y=-x+7 => 1

m=5/4, point = (9/2,6)
y=5/4x+b
6=9/2*5/4+b
6=45/8+b
b=3/8

y=5/4x+3/8 => 2

4) Set does equations equal to each other.
-x+7 = 5/4x+3/8
9/4x = 53/8
18x=53
x=53/18

5) Plug in the x-value and solve for y
y=-x+7
y=-53/18+7
y=73/18

6) The answer is...
(53/18, 71/18)