The Pythagorean theorem is a fundamental principle in geometry. It states that in a right-angled triangle, the square of the length of the hypotenuse equals the sum of the squares of the other two sides. If we call the two legs of the triangle a and b, and the hypotenuse c, then the formula is a squared plus b squared equals c squared. In this example, we have a triangle with sides 3, 4, and 5. The square of 3 is 9, the square of 4 is 16, and indeed, 9 plus 16 equals 25, which is the square of 5.
Let's look at a geometric proof of the Pythagorean theorem. We start with a square with side length a plus b. Inside this square, we can arrange four identical right triangles with legs of length a and b. The remaining space in the center forms another square. The side length of this inner square is c, which is the hypotenuse of our right triangles. The area of the outer square is (a plus b) squared, which equals a squared plus 2ab plus b squared. The area of the four triangles is 4 times one-half a times b, which equals 2ab. If we subtract the area of the triangles from the area of the outer square, we get a squared plus b squared. This must equal the area of the inner square, which is c squared. Therefore, c squared equals a squared plus b squared.
The Pythagorean theorem has numerous practical applications. One of the most common is finding the distance between two points in a coordinate system. For example, if we have two points A with coordinates (1,1) and B with coordinates (5,4), we can find the distance between them using the Pythagorean theorem. The horizontal distance, delta x, is 4, and the vertical distance, delta y, is 3. Using the distance formula, which is derived from the Pythagorean theorem, we calculate the square root of the sum of the squares of these differences. That's the square root of 16 plus 9, which equals 5. The theorem is also widely used in navigation, architecture, construction, engineering, and physics.
Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem. The most well-known example is the 3-4-5 triangle, where 3 squared plus 4 squared equals 5 squared. Other common examples include 5-12-13, 8-15-17, and 7-24-25. These triples are important in mathematics and have practical applications in construction and engineering. Interestingly, there's a formula to generate Pythagorean triples: if we take two positive integers m and n where m is greater than n, then a equals m squared minus n squared, b equals 2mn, and c equals m squared plus n squared. For example, with m equals 2 and n equals 1, we get the 3-4-5 triple.
To summarize what we've learned about the Pythagorean theorem: First, it states that in a right triangle, 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. Second, this theorem can be proven geometrically by comparing the areas of squares constructed on each side of the triangle. Third, the theorem forms the basis for calculating distances between points in coordinate systems. Fourth, Pythagorean triples are sets of three integers that satisfy the theorem, with the 3-4-5 triple being the most famous example. Finally, the Pythagorean theorem has numerous practical applications in fields such as navigation, construction, engineering, and physics, making it one of the most useful and important theorems in mathematics.