Find the area of this trapezoid!
don’t know the height : / that’s what’s tripping my up.

27

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Explanation:

See the attached image. I've plotted the points A, B, C, D and E as shown in the figure. Then I connected point B to point D
Rectangle ABDE attaches to triangle BDC at the shared segment BD. We don't know what BD is, so call it h for now

Notice how triangle BDC is a right triangle with the 90 degree angle at angle D. With this triangle, we know the horizontal leg is 4 units (ED+DC = EC; 7+4 = 11), the vertical leg is h, and the hypotenuse is 5. 

So basically
a = 4
b = h
c = 5

Using the pythagorean theorem, we find that
a^2 + b^2 = c^2
4^2 + h^2 = 5^2
16 + h^2 = 25
h^2 = 25-16
h^2 = 9
h = sqrt(9)
h = 3

The length of BD is 3 units. So the height is 3 units

From here, you can compute the area of the rectangle to get
A = L*W = 7*3 = 21

and the area of the triangle
A = b*h/2 = 4*3/2 = 12/2 = 6

Getting the answer of 21+6 = 27

Or you can use the trapezoid area formula to get the same result
A = h*(b1+b2)/2
A = 3*(7+11)/2
A = 3*18/2
A = 54/2
A = 27

Either way, the answer is 27