First off, are you sure it's ABC~EBF and not ADC~EBF? I don't think from the looks of things like ABCX is a triangle.

Assuming it is ADC~EBF, how about this for the first one below:

Since (1) AC cuts right through the middle of the rectangle, (2) triangle EBF is similar to triangle ADC, and (3) Triangle EBF touches triangle ADC and two sides of the rectangle, B is the center of the rectangle. That means the height of EBF would be half the height of ADC, and the width of EBF would be half the width of ADC. Since you're given the height and width of ADC, you can solve for the height and width of EBF. And once you know the height and width of EBF, use the pythagorean theorem to solve for the length of EF, the third side.

- Steve J