The Pythagorean Theorem is one of the most famous theorems in mathematics. It describes the relationship between the sides of a right triangle. In a right triangle, we have two legs labeled a and b, and the hypotenuse labeled c, which is the longest side opposite to the right angle.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides. This can be written as a squared plus b squared equals c squared. We can visualize this by drawing squares on each side of the triangle. The area of the square on the hypotenuse equals the sum of the areas of the squares on the two legs.
Let's solve a practical example using the Pythagorean theorem. If we have a right triangle where side a equals 3 and side b equals 4, what is the length of the hypotenuse c? We substitute into our formula: c squared equals a squared plus b squared. So c squared equals 3 squared plus 4 squared, which is 9 plus 16, equals 25. Taking the square root, we get c equals 5.
The Pythagorean theorem has countless real-world applications. In construction and architecture, it helps ensure buildings are square and calculate distances. For example, if you need to place a ladder against a wall, and the base is 3.5 feet from the wall while reaching 3.5 feet up the wall, you can calculate that the ladder needs to be approximately 4.95 feet long. GPS systems use it for navigation, computer graphics rely on it for distance calculations, and engineers use it in design work.