Proof of the Pythagorean Theorem

Why develop another proof of the Pythagorean theorem? Simple answer: because it stretches my mind. I think of it as mental yoga.

Pythagoras Statue — Image by Matthias Groeneveld from Pixabay

I worked this proof out not long after I retired. I have found that the best way for me to understand any mathematical concept is to attempt to prove it without reference to existing proofs. By doing this I learn where I may have gaps in my understanding. It also stretches my thinking, something that I hope will keep my brain active.

By working out my own proofs of a mathematical concept I quit simply being a consumer of other peoples ideas. I become a creator of my own ideas, insights, and connections. It doesn’t matter to me if something has already been done. It is still very exciting to me if I can work it out for myself!

The Pythagorean Theorem

Pythagoras’s theorem has been used in engineering for 4,000 years. The theorem says that if you have a right-angle triangle then the square of the hypotenuse is equal to the sum of the squares of the other two sides. Here’s a picture:

If a, b, and c represent the lengths of the sides of a right triangle, then written algebraically the theorem states that:

Mathematical fundamentals used in this proof:

• The sum the interior angles of any triangle on a plane is 180°.
• In a right triangle, the angle at the corner where a and b meet is 90° and therefore the sum of the other two angles is also 90°.
• The area of a right-angle triangle is 1/2 times the width of the base of the triangle times the height of the triangle.
• The order of mathematical operations is Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

Proof:

• Construct the following geometric form using the above triangle:

We can see that the area of the object would be length a times length b, or ab. Therefore the area of the triangle a,b,c would be:

• Now construct the following geometric form using the rectangle shown in the first step of the proof:

We can see within this object a square with sides of length c with each corner an angle of 90° which we know because it is equal to the sum of the two non-right-angle angles in each triangle.

• By eliminating the unnecessary extra triangles, we get the following geometric object:

• The area of the outer square can be calculated as:

• The area of the inner square can be calculated as:

• Note that the difference between the area of the two squares is equal to the area of four of the initial right-angle triangles. Each of these triangles has an area of:

• Therefore we now know that:

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This is just one of many proofs of this simple but highly useful relationship. The fun thing about retirement is that I finally have time to explore interests that the busyness and mental strain of working for a living overwhelmed. Now that I have the time I am exploring many neglected areas of interest. Over time I expect to share many of these explorations on Medium. If you found this article interesting, please let me know by clapping or by writing to me. I can be reached at wendell.rylander@gmail.com.

If you are interested in more proofs of the Pythagorean theorem, check out Algebraic Pythagorean Theorem 24 Proofs. (Note: I am no longer an Amazon Affiliate.)