The diagram attached consists of a rectangle ABCD and a triangle DXY so that X and Y are points on the line segments AB and BC respectively and angle DXY = 90 degrees. If all the line segments in the diagram have integer lengths, then we call it a Sophie Diagram.

a) A particular sophie diagram has AX = 12, XY = 15 and XD = 20. Find the length and width of rectangle ABCD.

b) Another sophie diagram has DX = 429 and DY = 845. Find the length and width of rectangle ABCD.