Question

What is the orthocenter of a triangle with corners at `(9 , 5 )`, `(3 , 8 )`, and `(5 ,6 )`?

Answer 1

Steps: (1) find the slopes of 2 sides, (2) find the slopes of the lines perpendicular to those sides, (3) find the equations of the lines with those slopes that pass through the opposite vertices, (4) find the point where those lines intersect, which is the orthocenter, in this case `(6.67, 2.67)`.

Explanation:

To find the orthocenter of a triangle we find the slopes (gradients) of two of its sides, then the equations of the lines perpendicular to those sides.

We can use those slopes plus the coordinates of the point opposite the relevant side to find the equations of the lines perpendicular to the sides that pass through the opposite angle: these are called the 'altitudes' for the sides.

Where the altitudes for two of the sides cross is the orthocenter (the altitude for the third side would also pass through this point).

Let's label our points to make it easier to refer to them:

Point A = `(9, 5)`
Point B = `(3, 8)`
Point C = `(5, 6)`

To find the slope, use the formula:

`m = (y_2-y_1)/(x_2-x_1)`

`m_(AB) = (8-5)/(9-3)=3/6=1/2`

`m_(BC) = (6-8)/(5-3)=(-2)/2=-1`

We don't want these slopes, though, but the slopes of the lines perpendicular (at right angles) to them. The line perpendicular to a line with slope `m` has slope `-1/m`, so the line perpendicular to `AB` has slope `-2` and the line perpendicular to `BC` has slope `1`.

Now we can find the equations of the altitudes of Point C (opposite AB) and Point A (opposite BC) respectively by substituting the coordinates of those points into the equation

`y=mx+c`

For Point C, the altitude is:

`6 = -2(5)+c` which gives `c=6+10=16` therefore `y = -2x + 16`

Similarly, for Point A:

`5=1(9)+c` which gives `c=5-9=-4` so the equation is:
`y=x-4`

To find the orthocenter, we simply need to find the point where these two lines cross. We can equate them to each other:

`-2x+16=x-4`

Rearranging, `3x=20 to x~~6.67`

Substitute into either equation to find the `y` value, which is `2.67`.

Therefore the orthocenter is the point `(6.67, 2.67)`.