The Pythagorean Theorem is one of the most famous theorems in mathematics. It states that in a right-angled triangle, the square of the hypotenuse equals the sum of squares of the other two sides. The formula is a squared plus b squared equals c squared, where a and b are the legs and c is the hypotenuse.
We can visualize the Pythagorean theorem by drawing squares on each side of the right triangle. The square on leg a has area a squared, the square on leg b has area b squared, and the square on the hypotenuse has area c squared. The theorem states that the sum of the areas of the two smaller squares equals the area of the largest square.
Let's verify the Pythagorean theorem with the famous three-four-five triangle. Given that a equals three and b equals four, we calculate a squared plus b squared equals nine plus sixteen, which equals twenty-five. Therefore, c squared equals twenty-five, so c equals five. This confirms that the three-four-five triangle is indeed a right triangle.
The Pythagorean theorem has countless real-world applications. It's used in construction and architecture to ensure structures are square and level. GPS systems use it for navigation calculations. Computer graphics rely on it for distance calculations. Engineers use it in design work. For example, to find the diagonal distance between two points, we use the formula d equals the square root of x squared plus y squared.
To summarize what we've learned about the Pythagorean Theorem: It applies only to right triangles and states that a squared plus b squared equals c squared, where c is the hypotenuse. We can prove it visually using square areas, and it has countless practical applications from construction to computer graphics. This fundamental theorem serves as the foundation for distance calculations throughout mathematics.