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 fundamental relationship is expressed as a squared plus b squared equals c squared, where c is the hypotenuse and a and b are the legs of the triangle.
To visualize the Pythagorean theorem, we construct squares on each side of the right triangle. The red square has area a squared, the green square has area b squared, and the yellow 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 work through a specific example. We have a right triangle with legs of length 3 and 4 units. Using the Pythagorean theorem, we calculate: 3 squared plus 4 squared equals c squared. This gives us 9 plus 16 equals c squared, which simplifies to 25 equals c squared. Taking the square root, we find that c equals 5. This is the famous 3-4-5 right triangle.
The Pythagorean theorem is essential in real-world applications. For example, if we need to place a ladder against a wall, and the base is 3 meters from the wall while reaching 4 meters high, we can calculate the required ladder length. Using the theorem, the square root of 3 squared plus 4 squared equals the square root of 25, which is 5 meters.