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'Folding Squares' printed from

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Take a $10$ cm square

Fold to create a diagonal.
Divide a side in half by folding.
Fold the paper again, joining the opposite corner to the midpoint of the halved side.
What fractions of the diagonal do you think your new fold has created?

Measure the two sections of the diagonal and compare their lengths to the diagonal's total length.
Is this what you expected?

Create or draw some more $10$cm squares and repeat the process.
Do you always end up with the same answer?

What fractions does the second fold appear to divide the diagonal into?

Does this appear to be the case for squares of different sizes?
Can you produce a convincing mathematical argument or proof that justifies what you have found?

Would the same work if you started with a rectangle or a parallelogram or a trapezium?
Justify your statements!