Question

What is the orthocenter of a triangle with corners at `(2 ,3 )`, `(5 ,1 )`, and (9 ,6 )#?

Answer 1

The Orthocenter is `(121/23, 9/23)`

Explanation:

Find the equation of the line that goes through the point `(2,3)` and is perpendicular to the line through the other two points:

`y - 3 = (9 - 5)/(1 -6)(x - 2)`

`y - 3 = (4)/(-5)(x - 2)`

`y - 3 = -4/5x + 8/5`

`y = -4/5x + 23/5`

Find the equation of the line that goes through the point `(9,6)` and is perpendicular to the line through the other two points:

`y - 6 = (5 - 2)/(3 - 1)(x - 9)`

`y - 6 = (3)/(2)(x - 9)`

`y - 6 = 3/2x - 27/2`

`y = 3/2x - 15/2`

The orthocenter is at the intersection of these two lines:

`y = -4/5x + 23/5`
`y = 3/2x - 15/2`

Because y = y, we set the right sides equal and solve for the x coordinate:

`3/2x - 15/2 = -4/5x + 23/5`

Multiply by 2:

`3x - 15 = -8/5x + 46/5`

Multiply by 5

`15x - 75 = -8x + 46`

`23x = + 121`

#x = 121/23

`y = 3/2(121/23) - 15/2`

`y = 3/2(121/23) - 15/2`

`y = 363/46 - 345/46`

`y = 9/23`

The Orthocenter is `(121/23, 9/23)`