Pythagorean Theorem and Triple Equations

The Pythagorean Theorem is a fundamental concept in mathematics that has played a significant role in shaping the field of geometry. It states that in any right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. This theorem has been widely studied and used by mathematicians and scientists for centuries, and it continues to be a crucial concept in modern mathematics.

Pythagorean Triple

A Pythagorean triple is a set of three positive integers that satisfy the Pythagorean Theorem. In other words, a, b, and c are considered a Pythagorean triple if a^2 + b^2 = c^2. Common examples of Pythagorean triples include (3, 4, 5), (5, 12, 13), and (8, 15, 17).

The Equation That Defines the Pythagorean Theorem

The Pythagorean Theorem can be expressed as an equation: a^2 + b^2 = c^2. This equation is used to calculate the length of a side in a right-angled triangle when the lengths of the other two sides are known.

Pythagorean Family

The Pythagorean family is a set of three Pythagorean triples that share a common side length. For example, (3, 4, 5), (6, 8, 10), and (9, 12, 15) are all members of the Pythagorean family that share the side length of 3.

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