Understanding the Pythagorean Theorem

Relation in Euclidean geometry among the three sides of a right triangle.

The Pythagorean theorem is one of the cornerstones of geometry, named after the ancient Greek mathematician Pythagoras. This theorem provides a relationship between the sides of a right-angled triangle. It states that "The square of the hypotenuse is equal to the sum of the squares of the other two sides". In mathematical terms, if a and b are the lengths of the two shorter sides, and c is the length of the hypotenuse, the theorem can be written as:

a² + b² = c²

Proof of the Pythagorean Theorem

There are many ways to prove the Pythagorean theorem. One of the simplest and most intuitive proofs is by rearrangement:

1. Start with a right-angled triangle and make a square with this triangle as one of its corners. The side of the square is a + b, and its area is (a + b)².

2. This square can be divided into two smaller squares (with areas a² and b²) and two copies of the original triangle.

3. Now, rearrange the two triangles and the square with area b² to form a larger square with side length c and area c².

4. It's clear that the total area of the square hasn't changed through this rearrangement. Therefore, a² + b² (the sum of the areas of the two smaller squares) must be equal to c² (the area of the larger square).

Practical Examples

Let's look at a practical example. Suppose we have a right-angled triangle with sides of lengths 3 and 4. We want to find the length of the hypotenuse.

Using the Pythagorean theorem, we can calculate:

3² + 4² = c²
9 + 16 = c²
25 = c²

So, c = √25 = 5. The length of the hypotenuse is 5.

The Pythagorean theorem is a powerful tool in geometry. It allows us to calculate distances, find missing lengths, and solve many practical problems. By understanding and applying this theorem, we can unlock a deeper understanding of the world around us.