2020 HMMT November General Round Problem 2

Let T be a trapezoid with two right angles and side lengths 4, 4, 5, and \sqrt{17}. Two line segments are drawn, connecting the midpoints of opposite sides of T and dividing T into 4 regions. If the difference between the areas of the largest and smallest of these regions is d, compute 240d.

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2020 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.