Two overlapping circles

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Question from Alexandra, a student:

There are two overlapping circles. The two non-overlapping regions have areas A and B.
As the area of overlap changes, the values of A and B also change.
Prove that no matter how big and small the overlap is, the difference between
A and B is always the same.
Guys please help!

Hi Alexandra,

Suppose the one circle has area $A_1$ square units and the non-overlapping part of this circle has area $A$ square units.

Suppose the other circle has area $B_1$ square units and the non-overlapping part of this circle has area $B$ square units.

How does $A - B$ compare to $A_1 - B_1?$

Penny

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