1 - The Backbone Theorem: The Pythagorean Theorem

Summary

Common Core State Standards

8.G.6 Explain a proof of the Pythagorean Theorem and its converse.

8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

What is the most famous theorem in all of mathematics?

Many would argue that the Pythagorean Theorem is the best known mathematical result. If you ask your friends and relations the above question I bet most will respond with this theorem. It is a fundamentally important result, key to many deep mathematical explorations, and this theorem will make many appearances throughout our thinking in this guide.

So let's start by being clear on what the theorem is and how to prove it.

The Pythagorean Theorem:Draw a square on each side of a right triangle (that is, a triangle with one angle of measure 90?) and label these squares I, II, and III as shown.

Then:

Area I + Area II = Area III

Two comments:

1. 1. I personally find this result very difficult to believe on a gut level! Look at the large square in the diagram. Does it look possible to you that its area really does equal the sum of the areas of the two smaller squares—on the nose?

2. 2. The Pythagorean Theorem is a statement from geometry. When asked to state the Pythagorean Theorem,most people rattle off “a squared plus b square equals c squared,” which sounds like a statement of algebra.

Of course, if we label the sides of the right triangle a, b, and c as shown, then Area I does indeed have value a², Area II value b², and Area III c². The statement “Area I +Area II =Area III” then translates to a² + b² = c².

But one need not label the sides of the right triangle with the letters a, b, and c.