Stephanie taught high school science and math and has a Master's Degree in Secondary Education.
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!
Trapezoids are 4-sided figures (quadrilaterals) with a pair of sides that are parallel and another pair that aren't. 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 either way and see that the distance between the lines stays the same.
In 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.
Let's refer to the two sides of the triangle that aren't parallel as the legs. The other two sides that are parallel will be called the bases.
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.
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.
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's 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.
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 2. 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 the 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.
This time, the lengths of the two bases are 8.975 cm and 3.4 cm. What is the length of the midsegment?
Start by adding up the two bases:
- 8.975 cm + 3.4 cm = 12.375 cm
Then divide the sum by 2:
- 12.375 cm / 2 = 6.1875 cm
Easy! The length of the midsegment is simply 6.1875 cm.
Now let's say you were given a problem where you were given:
- The length of the midsegment = 14
- The length of one of the bases = 16.5
How long is the other base? Can we use the theorem that way? Well, let's look at an example.
When solving problems like these, it's easiest to start with an equation and fill in what we know:
- Midsegment = (base1 + base2) / 2
Since we know the midsegment length, let's fill that in:
- 14 = (base1 + base2) / 2
Then we will fill in the base length that we know:
- 14 = (16.5 + base2) / 2
Now, we will use inverse operations to solve for base2. Start by multiplying both sides by 2:
- 28 = 16.5 + base2
Then subtract 16.5 from both sides to isolate base2:
- 11.5 = base2
Thus, the length of the second base is 11.5.
All right, let's take a brief moment to review. As we learned, trapezoids are 4-sided figures (quadrilaterals) with a pair of sides that are parallel and another pair that aren't. They have one pair of parallel sides, called bases, and another pair of non-parallel sides, called legs.
The midsegment of a trapezoid runs across the trapezoid starting at the midpoint of each leg, with the midpoint being equal distance from both ends of the lines of the legs. The length of the midsegment can be calculated using the trapezoid midsegment theorem, which states that the length of the midsegment is equal to the sum of the base lengths divided by 2. Therefore, it can be calculated by adding up the length of each base and dividing the sum by 2.
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