Pythagorean Theorem: Several Interesting Proofs and Facts

In this session, we will revisit Pythagorean Theorem. We will introduce several amazing proofs and some of its applications or fun facts.

• A few proofs

• Pythagorean Triples, which are integers a, b, c that satisfy $$a^2 + b^2 = c^2 $$, e,g, (3, 4, 5), (20, 21, 29), or (27007694075883, 41367460029844, x), ok. do you have a calculator? what is the integer x such that x^2 = 27007694075883 ^ 2 + 41367460029844 ^ 2. Did you know that Google search box can do math for you? Type this in the google 27007694075883 ^ 2 + 41367460029844 ^ 2 and find out the result. Unfortunately, it won t help you find out x.

• Now we will try to find all such triples. Yes, all. There are infinite of them.

• Unit circle is the circle of radius 1 centered at the origin. The points (x, y) on the circle has the property that x^2 + y^2 = 1. We will try to find such numbers so that x, y are both rational.

• Will talk about irrational numbers if the students do not have the concept.

• (Time permitting), will talk about some examples of ruler-and-compass construction.

• An example of constructing an irrational number

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In the meantime, I want to get some background from you. Could you take some time to answer the following? I want to plan the materials accordingly so that we can maximize the learning from this session.

• Do you know rational number v.s. irrational numbers?

• Have you done any proofs on some theorems in geometry?

• Do you know the concept of similar triangles and congruent triangles?

• Have you ever learned about ruler-and-compass construction? For example, given a line segment, construct an equilateral triangle with the given segment as a side using only ruler and compass.

• Have you learned about unit circle and some related mathematical properties associate with it?

• Do you know about Pythagorean Theorem before registering this session? If yes, please think about a proof of the Pythagorean Theorem or find one on the web and try to understand it.

Please let me know if you any questions or suggestions on the class.