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Trapezoid Midsegment Properties - Concept

Teacher/Instructor Brian McCall
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

A midsegment of a trapezoid is the line segment connecting the midpoints of the two non-parallel sides of a trapezoid. A trapezoid midsegment is parallel to the set of parallel lines in a trapezoid and is equal to the average of the lengths of the bases. A trapezoid midsegment is related to a triangle midsegment given that both of their lengths are proportional to the bases.

In a trapezoid where the bases are the 2 sides that are parallel we can draw a midsegment but how do we find a midsegment in a trapezoid? Well it's kind of similar to finding the midsegment in a triangle. First thing we're going to do is we're going to find one of our non parallel sides and we're going to find its midpoint. Then we're going to go over to the other non parallel side and find the midpoint of that segment. Then you're going to connect the 2 forming a line segment, so what I'm going to call this midsegment x I'm going to say it has the length of x there's 2 special things about this midsegment and a trapezoid. The first thing is, is that it is parallel to both of the bases so by finding the midpoint and connecting the midpoints of the we've created another parallel line.
The second key thing is that x this distance is equal to the average of the bases so the average means to add up and then divide by how many terms you have. So we only have 2 terms here, so I'm going to say x is equal to a plus b divided by 2. So if you're trying to find one of these missing sides but you know 2 of them all you have to do is plug them into this formula. But how is this similar to a triangle? Well we said in a triangle midsegment that x is equal to half of b so what's the relationship between a triangle midsegment and a trapezoid midsegment? Well if we looked at this trapezoid if I took this vertex right here and I dragged it all the way until it met this vertex right here.
Then what I would do is I'd be creating a triangle, so we could call this point a, so if we use the exact same formula where x the midsegment is the average of the bases we're going to end up with x equals one half b but why is that? Well as you can see a is just a point and points don't have any distance. So here a equals 0, so if I substitute it in 0 for this we're going to get x equals 0 plus b divided by 2 and 0 plus b is just b divided by 2. So that's the relationship between a triangle midsegment and a trapezoid midsegment.
Is that in the trapezoid we know that the midsegment will be parallel to the 2 bases and we could find its length by taking the average of the 2 bases.

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