The Pythagorean theorem is one of the most famous theorems in mathematics. It states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This relationship is expressed as a squared plus b squared equals c squared, where a and b are the legs and c is the hypotenuse.
To visualize why the Pythagorean theorem works, we can construct squares on each side of the right triangle. The square on side a has area a squared, the square on side b has area b squared, and the square on the hypotenuse c has area c squared. The visual proof shows that the sum of the areas of the two smaller squares equals the area of the largest square, confirming that a squared plus b squared equals c squared.
We can prove the Pythagorean theorem algebraically using a square arrangement. Start with a large square of side length a plus b, containing four identical right triangles and an inner square. The area of the large square equals the sum of the four triangles plus the inner square. This gives us the equation: a plus b squared equals four times one-half a b plus c squared. Expanding the left side gives a squared plus two a b plus b squared equals two a b plus c squared. Subtracting two a b from both sides yields our theorem: a squared plus b squared equals c squared.
The Pythagorean theorem has many practical applications. First, calculating TV screen diagonals: for a TV with width 48 inches and height 27 inches, the diagonal is the square root of 48 squared plus 27 squared, which equals 55.1 inches. Second, in construction for ensuring square corners: if two sides measure 3 feet and 4 feet, the diagonal should be 5 feet. Third, in navigation: if you travel 8 kilometers east and 6 kilometers north, your direct distance from the starting point is 10 kilometers.
Let's work through three examples. Example 1: Find the hypotenuse when a equals 3 and b equals 4. Using the formula, c equals the square root of 3 squared plus 4 squared, which equals 5. Example 2: Find a leg when the hypotenuse c equals 13 and b equals 5. Rearranging the formula, a equals the square root of 13 squared minus 5 squared, which equals 12. Example 3: A ladder problem. A 12-foot ladder leans against a wall with its base 5 feet from the wall. The height reached is the square root of 12 squared minus 5 squared, which equals 10.9 feet.