The bases of trapezoid $ABCD$ are $\overline{AB}$ and $\overline{CD}$. We are given that $CD = 8$, $AD = BC = 7$, and $BD = 9$. Find the area of the trapezoid.
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To find the area of a trapezoid, we can use the formula:
Area = $\dfrac{1}{2} \times$ (sum of the lengths of the bases) $\times$ (height)
In this case, the bases of the trapezoid are $\overline{AB}$ and $\overline{CD}$, and the height of the trapezoid is the perpendicular distance between the bases.
To find the height of the trapezoid, we need to consider the right triangle $\triangle ABD$.
Given that $AD = BC = 7$ and $BD = 9$, we can use the Pythagorean theorem to find the length of $\overline{AB}$.
Using the Pythagorean theorem, we have:
$(AB)^2 = (AD)^2 - (BD)^2$
$(AB)^2 = 7^2 - 9^2$
$(AB)^2 = 49 - 81$
$(AB)^2 = -32$
Since we cannot take the square root of a negative number, we know that there is no triangle with side lengths that satisfy the given conditions. Therefore, the trapezoid described in the problem does not exist.