As we noted in our previous problem of the week, “students’ development of mathematical reasoning involves making generalizations, and deeply felt generalizations are the result of making *many *specific observations…” So we offer a third approach to Pythagoras. Certain problems are also more compelling to some students than others.

This problem presents an additional feature that an irrational number can be a reasonable and even unavoidable answer to the length of a side or hypotenuse. Like the previous pythagorean problems, this one allows students to come at the solution by reasoning their way through a consideration of area. It involves students *thoroughly* noticing the given (the implications of the given information). For instance, what kind of triangle is triangle ADC? Do they stop with one observation or do they notice and consider the implications all of the given information about that triangle? And it enables them to carry forward and reapply previous knowledge like the properties of various triangles, a square, and the angle-side theorems.

So again, imagine that you had never heard of the Pythagorean Theorem (a^{2} + b^{2} = c^{2}), and that you did not have a ruler. Using only the drawing below, determine the length of line segment AC. All corners that look square are square. All lines that look parallel are parallel. The answer will be posted next week.

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