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Using the Trapezoid Midsegment Theorem

Instructor: Stephanie Matalone

Stephanie taught high school science and math and has a Master's Degree in Secondary Education.

In this lesson, we will discuss the basic definition of a trapezoid and its midsegment. Then, we will review the trapezoid midsegment theorem and go over some examples of how it is used.

So You Think You Can Trapezoid?

Are you a quadrilateral? Do you have a pair of parallel sides? Do you have another pair of sides that aren't parallel? If you answered yes to all of these questions, then our judges have determined that you are a trapezoid!

Trapezoid
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Trapezoids are four-sided figures (quadrilaterals) with a pair of sides that are parallel and another pair that are not. Parallel simply means that the sides are equal distance apart the whole way across, like train tracks. Starting at the middle, you can travel along them wither way and see that the distance between the lines stays the same!

In the a trapezoid, you can see that lines on the top and bottom of the trapezoid are parallel, while the lines on the sides are not parallel.

The lines in the image are parallel because they do not come together or drift apart at any point.
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Midsegment

Let's refer to the two sides of the triangle that are not parallel as the legs. The other two sides that are parallel will be called the bases.


Trapezoid legs and bases
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The midsegment of a trapezoid is a line that goes across from the middle of one leg to the middle of the other leg. The midsegment meets each leg at the midpoint. The midpoint is equal distance from both ends of the lines of the legs.


Midsegment and midpoints of a trapezoid
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On a Coordinate Plane

It is possible to use all of what we learned when we have a trapezoid on a coordinate plane. Let's say you are given the following four points:

  • (0,0) (5,0) (1,4) (5,4)

You can graph those on a coordinate plane, drawing lines between the points to find that it makes a trapezoid!


Trapezoid graphed with the coordinates (0,0) (5,0) (1,4) (5,4)
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This trapezoid looks a little different than the one we saw earlier, but you can still see that the bases are parallel and the legs are not. We can easily draw in our midsegment here. Remember that the midsegment must start at the midpoints of each leg.

Since both of our legs start at 0 on the y axis and end at 4 on the y axis, it is easy to see that the midpoints should fall at 2 on the y axis for both legs. Then we just draw our line across the see the midsegment!


Midsegment graphed
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Trapezoid Midsegment Theorem

Now that we understand some of the basics of trapezoids, let's talk about the trapezoid midsegment theorem, which states that the length of the midsegment is equal to the sum of the base lengths divided by two. In other words, the midsegment is the average length of the two bases.

So let's say we are looking at a trapezoid where the length of the base on top is 2 and the length of the base on bottom is 4. To find the length of the midsegment, we would simply add up the lengths of the bases:

  • 2 + 4 = 6

Then divide the sum by 2:

  • 6 / 2 = 3

Thus, the length of the midsegment is 3!

Using the Theorem

This time, the lengths of the two bases are 8.975 cm and 3.4 cm. What is the length of the midsegment?

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