Question

What is the orthocenter of a triangle with corners at `(4 ,9 )`, `(7 ,4 )`, and (8 ,1 )#?

Answer 1

Orthocenter: `(43,22)`

Explanation:

The orthocenter is the intersecting point for all the altitudes of the triangle. When given the three coordinates of a triangle, we can find equations for two of the altitudes, and then find where they intersect to get the orthocenter.

Let's call `color(red)((4,9)`, `color(blue)((7,4)`, and `color(green)((8,1)` coordinates `color(red)(A`,`color(blue)(B`, and `color(green)(C` respectively. We'll find equations for lines `color(crimson)(AB` and `color(cornflowerblue)(BC`. To find these equations, we'll need a point and a slope. (We'll use the point-slope formula).

Note: The slope of the altitude is perpendicular to the slope of the lines. The altitude will touch a line and the point that lies outside of the line.

First, let's tackle `color(crimson)(AB`:

Slope: `-1/({4-9}/{7-4})=3/5`

Point: `(8,1)`

Equation: `y-1=3/5(x-8)->color(crimson)(y=3/5(x-8)+1`

Then, let's find `color(cornflowerblue)(BC`:

Slope: `-1/({1-4}/{8-7})=1/3`

Point: `(4,9)`

Equation: `y-9=1/3(x-4)->color(cornflowerblue)(y=1/3(x-4)+9`

Now, we just set the equations equal to each other, and the solution would be the orthocenter.

`color(crimson)(3/5(x-8)+1)=color(cornflowerblue)(1/3(x-4)+9`

`(3x)/5-24/5+1=(x)/3-4/3+9`

`-24/5+1+4/3-9=(x)/3-(3x)/5`

`-72/15+15/15+20/15-135/15=(5x)/15-(9x)/15`

`-172/15=(-4x)/15`

`color(darkmagenta)(x=-172/15*-15/4=43`

Plug the `x`-value back into one of the original equations to get the y-coordinate.

`y=3/5(43-8)+1`
`y=3/5(35)+1`
`color(coral)(y=21+1=22`

Orthocenter: `(color(darkmagenta)(43),color(coral)(22))`