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
When learning about midpoints, it is also important to understand the concept of congruent segments. Congruent line segments are line segments with the same length. In a line segment, there is one point that will bisect the line segment into two congruent line segments. This point is called the midpoint.
If we're talking about the midpoint of a line segment which means that has fixed endpoints, we first have to talk about an idea of congruence. In Geometry congruence means the same distance or the same measurement. Before we filmed I drew two congruent segments. They measure exactly 9 inches. Notice that I use the two marks to denote inches. If I only used one of these, that would mean 9 feet but I'm looking at this, this is definitely not 9feet. So both of these are exactly 9 inches.
I'm going to add in endpoints that have labels. So I'm going to say this is line segment ab and I'm going to say that this is line segment cd. Now if I was going back to Algebra we love using equal signs in algebra. So I would say that ab=cd but now that we're in Geometry we're going to be using a different term and that different term is congruence. So I'm going to say that line segment ab notice I don't have arrows out here which tells me that it's not a ray and it's not a line, is congruent to line segment cd.
Now something that we should discuss right now is two lines can never be congruent because they're infinite. So in Algebra we'll say ab=cd in Geometry we say ab is congruent to cd. The only difference here is that we have Mr. Squiggles and I have the bar over ab which tells me we're talking about a line segment.
So getting back to our line segment, there is one point on this line segment that will bisect it. I'm going to write that down. Bisect is also a Geometry vocabulary word. Now what bisect means is that it will divide this into two congruent pieces. How can I tell every single Geometry student, every Geometry teacher, every mathematician in the world, that these two are congruent?
Well the way I do that is by using marks. So I'm going to write the same number of marks on either of these segments. If you wanted to, you could use two marks or if you're feeling crazy you could use three marks. But this tells every Geometry student that these that these two line segments have to be congruent.