The Pythagorean theorem is one of the most famous theorems in mathematics. It describes the relationship between the sides of a right triangle. The theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides, expressed as a squared plus b squared equals c squared.
Let's understand what each part of the formula means. In a squared plus b squared equals c squared, a and b represent the two legs of the right triangle, while c represents the hypotenuse, which is the longest side opposite the right angle. For example, if a equals 3 and b equals 4, then c squared equals 3 squared plus 4 squared, which is 9 plus 16 equals 25, so c equals 5.
The Pythagorean theorem can be proven visually by drawing squares on each side of the triangle. The area of each square equals the side length squared. For our 3-4-5 triangle, the red square has area 9, the green square has area 16, and the yellow square has area 25. Notice that 9 plus 16 equals 25, which confirms that a squared plus b squared equals c squared.
The Pythagorean theorem has many real-world applications. It's used in construction, navigation, computer graphics, and engineering. For example, to find the diagonal of a rectangular screen with width 16 inches and height 9 inches, we use the theorem: diagonal equals the square root of 16 squared plus 9 squared, which equals the square root of 256 plus 81, or approximately 18.4 inches.
To summarize what we've learned about the Pythagorean theorem: It applies only to right triangles and states that a squared plus b squared equals c squared, where c is the hypotenuse. The theorem can be proven visually using squares, has many practical applications, and serves as an essential foundation for geometry and trigonometry.