Question

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

Answer 1

`(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)`