Welcome to the Pythagorean theorem! This fundamental mathematical principle describes the relationship between the sides of a right-angled triangle. In any right triangle, the square of the hypotenuse equals the sum of squares of the other two sides.
Let's visualize the Pythagorean theorem by constructing squares on each side of our right triangle. The square on side a has area a-squared, the square on side b has area b-squared, and the square on the hypotenuse c has area c-squared. The theorem states that the area of the largest square equals the sum of the areas of the two smaller squares.
Let's apply the Pythagorean theorem to a real-world problem. We have a ladder that's 5 meters long leaning against a wall, with its bottom 3 meters from the wall. To find the height, we use our formula: h-squared plus 3-squared equals 5-squared. This gives us h-squared plus 9 equals 25, so h-squared equals 16, and therefore h equals 4 meters.
Pythagorean triples are special sets of three positive integers that perfectly satisfy the Pythagorean theorem. The most well-known triple is 3, 4, 5, where 3 squared plus 4 squared equals 5 squared. Other common triples include 5, 12, 13 and 8, 15, 17. These triples are useful in construction and engineering where exact right angles are needed.
The Pythagorean theorem has countless real-world applications. Architects use it to ensure buildings are square and stable. GPS systems rely on it for triangulation to determine your exact location. Computer graphics use it for calculating distances and rendering 3D objects. From ancient construction to modern technology, this simple formula a-squared plus b-squared equals c-squared continues to be one of the most useful tools in mathematics and science.