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 square of the hypotenuse equals the sum of the squares of the other two sides.
To use the Pythagorean theorem, we first need to identify the parts of a right triangle. The two shorter sides are called legs, labeled 'a' and 'b'. The longest side, opposite the right angle, is called the hypotenuse, labeled 'c'. The right angle is exactly 90 degrees.
Let's work through an example. We have a right triangle with legs of length 3 and 4. To find the hypotenuse, we apply the Pythagorean theorem. Three squared plus four squared equals c squared. Nine plus sixteen equals c squared. Twenty-five equals c squared. Therefore, c equals five.
Here's a visual proof of the Pythagorean theorem. We construct squares on each side of the triangle. The square on side 'a' has area 9, the square on side 'b' has area 16, and the square on the hypotenuse 'c' has area 25. Notice that 9 plus 16 equals 25, which proves that a squared plus b squared equals c squared.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides. This is written as a squared plus b squared equals c squared, where c is the hypotenuse and a and b are the legs.
In a right triangle, we have two legs and one hypotenuse. The legs are the two shorter sides that form the right angle. The hypotenuse is always the longest side and is opposite the right angle. In our example, the legs are 3 and 4 units, making the hypotenuse 5 units long.
We can prove the Pythagorean theorem using areas. When we construct squares on each side of the right triangle, the areas demonstrate the relationship. 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 our theorem.
Let's solve some examples. In the first problem, we know the legs are 3 and 4, and we need to find the hypotenuse. Using the formula, 3 squared plus 4 squared equals c squared. This gives us 9 plus 16 equals 25, so c equals 5. We can also solve for a missing leg when we know the other leg and hypotenuse.
The Pythagorean theorem has countless real-world applications. In construction, it helps determine ladder placement and building measurements. In navigation and GPS systems, it calculates distances between points. Computer graphics use it for rendering and animation. Engineers apply it in design and analysis. From simple distance calculations to complex physics problems, the Pythagorean theorem remains one of the most useful mathematical tools across science and technology.