Using Similarity to Prove the Pythagorean Theorem

The Pythagorean Theorem is one of the most interesting theorems for two reasons: First, it’s very elementary; even high school students know it by heart. Second, it has hundreds of proofs. The proof below uses triangle similarity.

Pythagorean Theorem

In a right triangle with side lengths a and b and hypotenuse c,  the following equation always holds:

c^2 = a^2 + b^2. » Read more

Generating Pythagorean Triples from Square Numbers

A figurate number is a number that can be represented by a regular geometrical arrangement of equally spaced points (or circles as shown in the first figure). If the arrangement forms a regular polygon, the number is called a polygonal number.

Examples of polygonal numbers are square numbers. The first  four square numbers are 1, 4, 9, and 16, and their geometric representations are shown in the first figure. It is clear that that the 10th square number has 102 circles, and in general, the nth square number has n2 circles.

Looking at the color pattern above, we can see that there is something very special about square numbers. Each square number can be represented as the sum of odd integers.  The first four examples are shown below. » Read more

Pythagorean Triple 2: Generating Pythagorean Triples

In the previous math article, we have shown that there are infinitely many Pythagorean triples.  In this article, we are going to discuss a very short but effective strategy in generating Pythagorean Triples.

A Pythagorean triple is the integer triple (a,b, c) satisfying the Pythagorean equation a^2 + b^2 = c^2.

Observe the Pythagorean triples (3, 4, 5) and  (5, 12, 13).  We can see that the hypotenuse is greater than the longer side by 1. From the pattern, we can form the Pythagorean triples (a, b, b + 1) satisfying the equation a^2 + b^2 = (b + 1)^2.

Right triangle with side length 3, 4 and 5 units.

Solving the equation we have a^2 = 2b + 1, which implies that  a = \sqrt{2b +1}.  Now,   2b + 1 is always odd (can you see why?). It follows that in order for a to be an integer, 2b + 1 must be a perfect square. This means, that we are sure that a is an integer, if 2b + 1 is an odd perfect square.

From here, we can generate infinitely many examples of Pythagorean triples.  For example, 49 is an odd perfect square. So plugging it in the equation we have,  2b + 1 = 49, then, we have the triple (7, 24, 25), another Pythagorean Triple.  If we let 2b + 1 = 121, then we have the triple (11, 60, 61).

Now, we found another way to generate Pythagorean Triples.

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