In middle school students studied the Pythagorean Theorem, including proofs of the Pythagorean Theorem. The proofs students studied in middle school were likely based on decomposing and rearranging the squares on each hypotenuse of the right triangle. An example of such a proof is included in this lesson.
In this lesson, students learn another way to prove the Pythagorean Theorem based on the fact that right triangles (and only right triangles) can be decomposed into two similar versions of themselves. The proof students study and complete is based on using the proportional relationships among the side lengths of the triangles to write expressions for \(c^2\) in terms of \(a^2\) and \(b^2\). Students must look for structures (MP7) as they figure out how to rewrite expressions to show that \(c^2 = a^2 + b^2\). They also get a chance to construct arguments (MP3) as they figure out how to turn their findings into a convincing argument that the Pythagorean Theorem is true.
Re-proving something they’ve already studied one or more proofs of also gives students a chance to reflect on what mathematicians do. Why might mathematicians be interested in multiple proofs of the same fact? What different unique things about right triangles do the different proofs reveal? How does each proof show the creativity of the people who came up with it? Could there be more ways of proving the Pythagorean Theorem that have not been discovered?
- Prove the Pythagorean Theorem (using words and other representations).
- Let’s prove the Pythagorean Theorem.
- I can prove the Pythagorean Theorem.
An altitude in a triangle is a line segment from a vertex to the opposite side that is perpendicular to that side.