Question

What is the orthocenter of a triangle with corners at `(5 ,4 )`, `(2 ,3 )`, and (7 ,8 )#?

Answer 1

The orthocenter is `=(10,-1)`

Explanation:

Let the triangle `DeltaABC` be

`A=(5,4)`

`B=(2,3)`

`C=(7,8)`

The slope of the line `BC` is `=(8-3)/(7-2)=5/5=1`

The slope of the line perpendicular to `BC` is `=-1`

The equation of the line through `A` and perpendicular to `BC` is

`y-4=-1(x-5)`

`y-4=-x+5`

`y+x=9`...................`(1)`

The slope of the line `AB` is `=(3-4)/(2-5)=-1/-3=1/3`

The slope of the line perpendicular to `AB` is `=-3`

The equation of the line through `C` and perpendicular to `AB` is

`y-8=-3(x-7)`

`y-8=-3x+21`

`y+3x=29`...................`(2)`

Solving for `x` and `y` in equations `(1)` and `(2)`

`y+3(9-y)=29`

`y+27-3y=29`

`-2y=29-27=2`

`y=-2/2=-1`

`x=9-y=9+1=10`

The orthocenter of the triangle is `=(10,-1)`