Trigonometry/Exercise: A Puzzle Triangle

What is wrong with the above figure? What is the area of each triangular piece, and what is the area of the whole triangle? Are both diagrams using the same pieces? What are the lengths of the hypotenuses of the two smaller triangles and of the whole large puzzle triangle(s)? (Use the Pythagorean Theorem) How would you explain to someone else what is going on?

This puzzle is telling us something important about proof.

• Could the first proof of the Pythagorean theorem be 'wrong' in the same way? Do the pieces really fit together the way they appear to do?

The first proof of the Pythagorean Theorem is OK, but it turns out that to really check that diagrammatic proof properly you need to have first proved that the angles of a triangle add to 180 degrees. That will be our next proof.