Welcome to the Pythagorean Theorem! This fundamental theorem in geometry describes the relationship between the sides of a right triangle. In any right triangle, the square of the hypotenuse equals the sum of squares of the other two sides. The formula is a squared plus b squared equals c squared, where c is the hypotenuse and a and b are the legs.
To use the Pythagorean Theorem, we must first identify the components of a right triangle. First, locate the right angle, which measures exactly 90 degrees. Next, find the hypotenuse - this is always the longest side and sits opposite the right angle. Finally, identify the two legs, which are the remaining sides that form the right angle. Only right triangles have this special relationship between their sides.
Now let's apply the Pythagorean formula with a concrete example. We have a right triangle with legs of length 3 and 4. Using the formula a squared plus b squared equals c squared, we substitute our values. Three squared equals 9, and four squared equals 16. Adding these together gives us 25. Taking the square root of 25 gives us 5. Therefore, the hypotenuse has length 5. This is the famous 3-4-5 triangle, one of the most common Pythagorean triples.
Here's a visual proof of the Pythagorean Theorem. We construct squares on each side of our right triangle. The square on leg a has area 9, the square on leg b has area 16, and the square on the hypotenuse has area 25. The theorem tells us that the sum of the areas of the two smaller squares equals the area of the largest square. Indeed, 9 plus 16 equals 25, confirming our theorem. This geometric relationship has been known for thousands of years and forms the foundation for many mathematical concepts.