Rectangle abcd and right triangle dce have the same area. they are joined to form a trapezoid, as shown. what is de?

Since the areas are the same  (5 x 6 = 30 units), and the height of the triangle (DC) is 5 so the length of the triangle must be 12.  (1/2 of  5 x 12 is also 30)

So now we use the Pythagorean theorem

a² + b² = c²   we have 5² + 12² = c²  or 25 + 144 = c²  or 169 = c²

The square root of 169 is 13 so side DE is 13 units.