The Pythagorean theorem is one of the most famous theorems in mathematics. It states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Here we see a right triangle with sides a, b, and c, where c is the hypotenuse.
The Pythagorean theorem is expressed by the formula a squared plus b squared equals c squared. Here, a and b represent the lengths of the two legs, while c represents the hypotenuse. Let's verify this with a 3-4-5 triangle: 3 squared plus 4 squared equals 9 plus 16, which equals 25, and 25 equals 5 squared.
We can prove the Pythagorean theorem visually by constructing squares on each side of the right triangle. The area of each square equals the square of its side length. The red square has area 9, the green square has area 16, and the yellow square has area 25. Notice that 9 plus 16 equals 25, confirming that the area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides.
The Pythagorean theorem has countless practical applications in our daily lives. It's used in construction, navigation systems, computer graphics, and engineering. For example, to find the distance between two points on a coordinate plane, we can use the theorem. Here we calculate the distance between points (0,0) and (6,8). We form a right triangle and apply the formula: distance equals the square root of 6 squared plus 8 squared, which equals the square root of 100, giving us a distance of 10 units.
In summary, the Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This fundamental relationship, expressed as a squared plus b squared equals c squared, only applies to right triangles where c is the hypotenuse. The theorem has countless real-world applications and serves as the foundation for distance formulas in mathematics. It beautifully connects geometry with algebra and remains one of the most important mathematical discoveries in human history.