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
A median is a line segment from the vertex to the midpoint of the opposite side in a triangle. In every type of triangle, the median will be contained within the polygon, unlike altitudes which can lie outside the triangle. When constructing a median, we first find the midpoint of the side opposite the desired vertex, then use a straightedge to connect the midpoint and the vertex.
A special segment in every triangle is a median and there's three of them because you can draw one from each vertex. The definition is that it's a line segment from one of those vertices to the midpoint of the opposite side.
So let's take a look at three special opposite cases, one for acute, right and obtuse. So if I picked this vertex, the first thing that you have to do is find the midpoint of the side that is opposite. So that midpoint will divide it into two congruent segments and the median will be a line segment connecting to that midpoint. So this is going to happen for all three of your sides. Notice that all of these medians are going to be within your triangle for an acute triangle.
Let's look at a right triangle. So we have to assume that that's going to be your right angle and if you find the midpoint of your opposite side here, we see that that median is within our triangle. If I find the midpoint there we see that this median will be inside our triangle and of our third side, we see that that median will be inside your triangle. So acute triangles, medians are all inside, right triangles medians are all inside.
Let's check obtuse triangles. So if I find the midpoint of this side, and if I draw a median from that vertex, if I find the midpoint of that side and I find the median and of my third side find the midpoint, and notice I'm using a different number of markings, and if I draw in my median. I see that no matter what kind of triangle we have, obtuse, right or acute, our medians will be within that triangle.