What is the orthocenter of a triangle with corners at `(3 ,1 )`, `(1 ,3 )`, and (5 ,2 )#?

Let's begin by writing the equation of the line that goes through point `(3,1)` and perpendicular to the line going through points `(1,3)` and `(5,2)`.

The slope, m, of the line going through points `(1,3)` and `(5,2)` is:

`m = (3 - 2)/(1 - 5) = -1/4`

The slope, n, of any line perpendicular is

`n = -1/m = 4`

Use point slope form of the equation of a line to obtain the equation of a line that we desire:

`y - y_1 = n(x - x_1)`

`y - 1 = 4(x - 3)`

`y - 1 = 4x - 12`

`y = 4x - 11`

Write the equation of a line that goes through point `(5,2)` perpendicular to the line through the points `(3,1)` and `(1,3)`:

The slope, m, of the line that goes through the points `(3,1)` and `(1,3)` is:

`m = (3 - 1)/(1 - 3) = 2/-2 = -1`

The slope, n, of any line perpendicular is:

`n = -1/m = 1`

Again, Use the point slope form for the point `(5,2)`:

`y - y_1 = n(x - x_1)`

`y - 2 = 1(x - 5)`

`y = x - 7`

The orthocenter is at the intersection of these two lines:

`y = 4x - 11`
`y = x - 7`

`x - 7 = 4x - 11`

`4 = 3x`

`x = 4/3`

`y = 4/3 - 7`

`y = -17/3`

The orthocenter is at the point `(4/3, -17/3)`