Tricky Trapezoid!

Draw an altitude from B to DC; name the intersection E. The distance BE is 80 since it is the opposite side of a rectangle.

Using the Pythagorean Theorem, EC = $\sqrt{89^2-80^2}=39$ . This means ED, and thus the opposite side of the rectangle AB, both have length (123 - 39) or 84.

Now in triangle ABD, use the Pythagorean Theorem again to find BD. $\sqrt{80^2+84^2}=116$ .