5.05 Trapezoids

Example #3

Verify that EF is the midsegment of the trapezoid.

Trapazoid A B C D has points A (-2, 0), B (0, 4), C (2, 4) and D (8, 0). Midpoints E at (-1, 2) and F at (5, 2)

Don't get thrown off by the coordinate plane. This simply means that we can use the coordinates of the vertices and formulas to find missing information. Remember, for EF to be the midsegment, we have to meet three conditions.

E and F are the midpoints of AB and DC
EF is parallel to AD and BC
EF is half the sum of BC and AD

Let's start by verifying that E and F are the midpoints of AB and DC. We'll use the midpoint formula to check that the coordinates of E and F are the midpoints of AB and DC.

Let's start with E(-1, 2).

Same as previous image. Trapazoid A B C D has points A (-2, 0), B (0, 4), C (2, 4) and D (8, 0). Midpoints E at (-1, 2) and F at (5, 2)

midpoint This is the formula for midpoint. = Apply midpoint theorem. X value = average of x1 and x2. Y value = average of y2 and y1

E Substitute A(-2,0) for (x₁, y₁) and B(0,4) for (x₂, y₂) = x value is (-2 + 0) over 2 and y value is (0 + 4) over 2

E Simplify the numerators by adding = x value is -2 over 2 and y value is 4 over 2

E = (-1, 2)

Now, verify that F(5, 2) is a midpoint.

F Substitute D(8,0) for (x₁, y₁) and C(0,4) for (x₂, y₂) = x value = (8 + 2) over 2 and y value = (0 + 4) over 2

F Simplify the numerators by adding = x value = 10 over 2 and y value equals 4 over 2

F Divide to simplify the fractions = (5, 2)

Ok, now that we have shown that E and F are the midpoints of AB and DC, let's see if EF is parallel to AD and BC. We'll start by finding the slope of AD, which will be equal to BC since they are parallel. Usa A as (x₁, y₁) and D as (x₂ y₂).

Slope = formula for slope. (y1 minus y2) over (x2 minus x1)

Slope = (0 minus 0) over (8 minus -2)

Slope =

Slope = 0

Now, let's find the slope of EF. If it is zero, than we know that EF is parallel to AD and BC. Usa E as (x₁, y₁) and D as (x₂ y₂).

Slope EF = formula for slope. (y1 minus y2) over (x2 minus x1)

Slope EF = (2 minus 2) over (5 minus -1)

Slope EF =

Slope EF = 0

Ok, the last thing to verify is that EF is half the sum of BC and AD. To find this, we'll need to first calculate the lengths of all of the segments. Because these lines have a slope of 0, we can count in this case. In others, you would need to use the distnace formula.

BC = ___ EF = ___ AD = ___

Answer: 2, 6, 10

Knowing that BC = 2, EF = 6, and AD = 10, we can now use the midsegment formula to see if EF is half the sum of BC and AD.

E F = (A D + B C) over 2

E F = (10 + 2) over 2

EF = 12 over 2

EF = 6

Since E and F are midpoints of the legs, EF is parallel to both bases, and EF is one half the sum of the lengths of AD and BC, it is verified to be the midsegment.