Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
Altitudes are defined as perpendicular line segments from the vertex to the line containing the opposite side. In each triangle, there are three triangle altitudes, one from each vertex. In an acute triangle, all altitudes lie within the triangle. In a right triangle, the altitude for two of the vertices are the sides of the triangle. In an obtuse triangle, the altitude from the largest angle is outside of the triangle.
When we're talking about triangles, there's a special segment three in each triangle called an "Altitude." So we're not talking about skiing here. What we're talking about is a perpendicular segment, remember this symbol right here means perpendicular-I'm trying to get you used to seeing these symbols-from a vertex to a line containing the opposite side.
So this definition is written very carefully. It's not always to the opposite side and you're going to see why in a second here. So if we look at an acute triangle, there are going to be three altitudes, one form each vertex.
So if I were to pick this top vertex right here, the altitude would go straight down perpendicular to the opposite side. We would have two more altitudes, each of which would go perpendicular to the opposite side. Notice that all three altitudes are inside the triangle.
If we look at a right triangle over here we can see that if I pick this vertex right here, we already have an altitude drawn. That's going to be that leg of the triangle. If I pick this vertex right here the altitude will just be that leg of the triangle. However if I pick my 90 degree angle as my vertex, then we'll be able to see that altitude inside the triangle.
So a third case is the obtuse triangle, and here is where I say to a line containing the opposite side. So if we pick this vertex, our opposite sides are over here but that opposite side doesn't continue to where this altitude will drop. Notice that I had to extend that opposite side.
If I look at the other two altitudes in this obtuse triangles, we're going to have one altitude going like that I'm going to have to extend that side as well and we'll drop down another altitude. So it's okay to have an altitude that is not inside your triangle.