1. ## Geometry help needed! Altitude, Orthocenter, and Circumcenter? Please help me out!

I'm really having some trouble finding Altitude, Circumcenter, and Orthocenter and I have a test on it tomorrow.

Let's say triangle ABC, A(-3,6) B(-9,-3) C(4,2). I kind of understand the concept..but kind of not really. =/

Thanks much!

2. Are you working out of the blue Discovering Geometry book?

To start with, an altitude goes from one vertex to the (infinite) line of the far side at a right (perpendicular) angle.

So to find an altitude, we'll need to:
1.) Find the slope of the far side.
2.) Convert that to a perpendicular (opposite reciprocal) slope for the altitude.
3.) Take our new slope, and the vertex our altitude passes through, and make an equation of a line.

For example, if you wanted the altitude from B:

1.) The slope of the far side is $\displaystyle \frac{2-6}{4-(-3)} = \frac{-4}{7}$

2.) The slope perpendicular to $\displaystyle \frac{-4}{7}$ is $\displaystyle \frac{7}{4}$

3.) So our new slope is $\displaystyle \frac{7}{4}$, and our vertex is B(-9,-3), so our equation is:

$\displaystyle y - (-3) = \frac{7}{4}(x - (-9))$

$\displaystyle y + 3 = \frac{7}{4}x + \frac{63}{4}$

$\displaystyle y = \frac{7}{4}x + \frac{51}{4}$

3. An orthocenter is the intersection of all three altitudes. To find it, you'll need to find two of the altitudes using the process above, and find the point where they intersect. I don't know your teacher, but it's usually a good idea to find the third altitude and check if your point of intersection is on the third altitude, as well. The reason I recommend that is because this is a lot of algebra, and one small mistake early on can screw you up without being noticeable at all.

4. Finally, the circumcenter is the intersection of all three perpendicular bisectors, so it'll help to run through the process of finding one of those. To find the perpendicular bisector of any one side, you must:
1.) Find the slope of the side you want to perpendicularly bisect.
2.) Convert that to a perpendicular (opposite reciprocal) slope for the perpendicular bisector.
3.) Find the midpoint of the side (this is how it will bisect).
4.)Use your new slope and the midpoint to make an equation of a line.

Let's say you wanted the perpendicular bisector of side AC.

1.) The slope of that side is $\displaystyle \frac{2-6}{4-(-3)} = \frac{-4}{7}$.

2.)
The slope perpendicular to is

3.) The midpoint of AC is $\displaystyle (\frac{-3+4}{2}, \frac{6+2}{2}) = (\frac{1}{2}, 4)$

4.) Our equation is:

$\displaystyle y - 4 = \frac{7}{4}(x - \frac{1}{2})$

$\displaystyle y - 4 = \frac{7}{4}x - \frac{7}{8}$

$\displaystyle y = \frac{7}{4}x +\frac{25}{8}$

Now, find another perpendicular bisector, find the intersection of that equation and this one, and again, I recommend trying that intersection in the equation of the third perpendicular bisector.