Answer: the diagonals of a trapezoid do not bisect each other.
Explanation:
We're going to prove that the diagonals of a trapezoid do not bisect each other. But instead of just straight up telling you the proof, we're going to do it using something called an indirect proof. This means we're going to assume that the diagonals do bisect each other and then show that this leads to a contradiction.
So, let's assume that the diagonals of our trapezoid, call them d1 and d2, do bisect each other. That means that they cut each other in half. Now, since a trapezoid has exactly one pair of parallel sides, we know that the midpoint of d1 and d2 must also be the midpoint of the parallel sides.
But here's the catch: the midpoint of a line segment is the same distance from both ends of the segment. So, if the midpoint of d1 is the same as the midpoint of d2, that means that d1 and d2 are the same length! And that's a contradiction, because we know that the diagonals of a trapezoid are not the same length.
So, by assuming that the diagonals of a trapezoid bisect each other, we've shown that it leads to a contradiction. And that means that our assumption was wrong, which means that the diagonals of a trapezoid do not bisect each other.