Brian McCall

**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

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A **midsegment** in a triangle connects the midpoints of two sides. This is different from a **median**, which connects a vertex to the midpoint of the opposite side. To construct a midsegment, find the midpoint of two sides. This can be done by drawing a perpendicular bisector on one side of the triangle. The point at which the bisector intersects the side is the midpoint of that side. Repeat this with the second side. Then, by connecting these two midpoints, a midsegment is created.

As a geometry student there are so many vocabulary words flowing around. It’s pretty easy to confuse midsegment with a median. So if we read this problem it says construct midsegment AB in triangle DEF where A is opposite of F and B is opposite of D. So how do I know if I forgot what a median and what a midsegment is?

Well the first clue is that we have points A and B and I don’t see A and B in this triangle, so I know that I’m creating two new points as my endpoint. A median will contain the vertex, the midsegment will not.

So if we were to sketch our game plan over here. So we’ve got our triangle DE and F. We said that A is opposite of F, so I know that A is going to be over here and that’s going to be the midpoint and that B is opposite of D, it says right at the bottom here. So I’m going to find point B by bisecting this line segment EF, but we’re not done. We have to draw the line segment that connects them. So that’s our game plan bisect one side, bisect another side connect the midpoints.

So to do that we’re going to need a compass and it doesn’t matter which side you start with, I’m going to start with side DE and in order to have enough room over here, I’m going to erase our game plan. We know that we just need to bisect two sides, so I’m going to swing an arc from vertex D, keeping my compass the same swing the same arc from vertex E. So now I’m going to take my straightedge and find that midpoint.

So if I were to draw in this line right here, it would be the perpendicular bisector of DE, but that’s not what we’re looking for, all we need to know is the mid-point. So that point right we’re going to call point A because it has bisected that side DE we need to do this one more time.

Come over to side EF, then I’m going to swing another arc from vertex F keep your compass the same swing an arc from point E and again we’re going to use our straightedge to find the midpoint of this side EF. So I’m using these two points of intersection. The two points where our arcs crossed and I have my midpoint B right there which has bisected our side EF.

Now to construct the mid-segment I need to connect the two midpoints, so I’m going to use my straightedge to do that and we’ve constructed our midsegment which we said is the line segment that connects the two midpoints in a triangle.