
1 Attachment(s)
Sophie Diagrams
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.

Hello, gunsandroses234!
In right triangle
In right triangle
Let: .
In right triangle .[1]
In right triangle
. .
We have: .
Substitute into [1]: .
. . . . . . . . . . . . . .
Therefore: .

I may be wrong, but I don't think that is correct.
Pythagoras gives you that .
We now want to find AB. However, there is only one pythagorean triple with hypotenuse 15, (9, 12, 15). We therefore need to find out if BY is 9 or 12. This is equivalent to showing whether AB=CD=12+9=21 or 12+12=24
To do this, note that DY = 25 by Pythagoras (the triple (15, 20, 25) ) and note that there are only two triples with hypotenuse 25,
(7, 24, 25) and (15, 20, 25).
Clearly, as 21 is not in either of these triples AB=CD=24, and we are done.
AB=24, AD=16.