The Pythagorean theorem is one of the most famous principles in mathematics. It describes the relationship between the sides of a right-angled triangle. In any right triangle, we have two shorter sides called legs, labeled a and b, and the longest side called the hypotenuse, labeled c, which is always opposite the right angle.
The Pythagorean theorem can be expressed with the famous formula: a squared plus b squared equals c squared. This means that if you take the square of each leg length and add them together, the result equals the square of the hypotenuse length. The squares shown here represent the areas of squares built on each side of the triangle.
Let's work through a practical example. We have a right triangle with legs of length 3 and 4 units. To find the hypotenuse, we use the Pythagorean theorem. 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, we find that c equals 5 units.
The Pythagorean theorem has countless real-world applications. It's used in construction and architecture, navigation systems, computer graphics, and engineering design. For example, if you need to find the diagonal distance across a rectangular field that is 60 meters by 80 meters, you can use the theorem. The diagonal equals the square root of 60 squared plus 80 squared, which gives us 100 meters.
To summarize what we've learned about the Pythagorean theorem: It applies only to right triangles and uses the formula a squared plus b squared equals c squared. This powerful tool helps us calculate unknown side lengths and has countless applications in science, engineering, and everyday problem-solving. It truly is one of the most fundamental and useful principles in mathematics.