The Pythagorean theorem is one of the most famous theorems in mathematics. It tells us that in any right triangle, if we square the lengths of the two shorter sides and add them together, we get the square of the longest side, called the hypotenuse. This relationship can be written as a squared plus b squared equals c squared.
Let's look at the formula more closely. We write it as a squared plus b squared equals c squared. Here, a and b represent the lengths of the two legs of the right triangle - these are the sides that form the right angle. The letter c represents the hypotenuse, which is always the longest side and sits opposite the right angle. This simple equation captures a fundamental relationship in geometry.
Let's work through a classic example. We have a right triangle where one leg is 3 units long and the other leg is 4 units long. We want to find the length of the hypotenuse. Using our formula, we substitute the values: 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 of both sides, we get c equals 5. This 3-4-5 triangle is one of the most famous right triangles.
Here's a beautiful visual proof of the Pythagorean theorem. If we construct squares on each side of our right triangle, we can see the relationship clearly. The red square on the leg of length 3 has an area of 9. The green square on the leg of length 4 has an area of 16. The yellow square on the hypotenuse of length 5 has an area of 25. Notice that 9 plus 16 equals 25, which confirms our theorem visually.
The Pythagorean theorem has countless real-world applications. Architects use it to ensure buildings are square and stable. GPS systems use it for distance calculations. Computer graphics rely on it for 3D rendering. Engineers apply it in design and construction. Remember the key points: it only works for right triangles, c is always the longest side opposite the right angle, and the formula is a squared plus b squared equals c squared. This ancient theorem continues to be one of the most useful tools in mathematics and science.