Impossible Picture?

Age 14 to 16 Challenge Level:

Take a look at the picture below:

It looks like you can dissect a square into 6 pieces, and reassemble it into 3 new squares:



In practice, this only works when the dissection has particular proportions.
Can you describe when it is possible?

Click below to reveal some hints.

Use this interactive to see how the pieces could change.

Can you describe when the orange parts would make a square?


 
  
You might need to use Pythagoras' Theorem.

You might need to express your answer using surds.



What is the ratio of the areas of the 3 squares?



With thanks to Don Steward, whose ideas formed the basis of this problem.