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# Trapezoid

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One base of a trapezoid is four times as long as the other. The height is the average of the two bases. If the area of the trapezoid is 100yd^2, find the length of the longer base.

Sep 1, 2021

#1
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Glad this isn't a question of trapezoid perimeter, those are a complete nightmare.

Anyways, let's start by making systems of equations from what we get in the problem.

$$B_L = 4B_S \\ H = (B_L + B_S) \div 2 \\ A = 100$$

This is what we can get from the description of the problem. Now let's apply what we know about trapezoids to add the important equation that puts it all together:

$$A = \frac{B_L + B_S}{2} \times H$$

"The area of a trapezoid is equal to the average of its bases times its height."

Looks a bit like the second equation above, right?
Let's use some reverse substitution:

$$A = H \times H \\ 100 = H^2 \\ H = 10$$

Now, time for more substitution to solve for the smaller base, as it will result in less shenanigans.

$$10 = (B_L + B_S) \div 2 \\ 20 = 4B_S + B_S \\ 20 = 5B_S \\ B_S = 4$$

And now we solve for the long base:

B_L = 4*4
B_L = 16

Thus, the length of the long base is 16 yards.

Sep 2, 2021