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, the side opposite the right angle is called the hypotenuse, and the other two sides are called legs.
The Pythagorean theorem states that a squared plus b squared equals c squared, where a and b are the lengths of the legs, and c is the length of the hypotenuse. For example, in a triangle with legs of length 3 and 4, the hypotenuse has length 5, because 3 squared plus 4 squared equals 9 plus 16, which equals 25, and 5 squared also equals 25.
We can prove the Pythagorean theorem visually by constructing squares on each side of the right triangle. The area of the square on the hypotenuse equals the sum of the areas of the squares on the two legs. In our example, the square on side a has area 9, the square on side b has area 16, and the square on the hypotenuse has area 25. Indeed, 9 plus 16 equals 25, confirming the theorem.
The Pythagorean theorem has many practical applications in real life. In construction and architecture, it helps find diagonal distances. In navigation and GPS systems, it calculates shortest paths. Engineers use it to determine cable lengths, and computer graphics rely on it for distance calculations. For example, if a ladder is placed 3 meters from a wall and reaches 4 meters high, the ladder itself must be 5 meters long, as shown by the formula 3 squared plus 4 squared equals 5 squared.
To summarize what we have 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. The theorem can be proven visually using squares constructed on each side, and it has countless practical applications in construction, navigation, engineering, and computer graphics. The Pythagorean theorem remains one of the most important and useful theorems in mathematics.