The Pythagorean theorem is one of the most famous theorems in mathematics. It describes a fundamental relationship in right-angled triangles. The theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides. This can be written as a squared plus b squared equals c squared, where c is the hypotenuse.
Now let's prove the Pythagorean theorem using a geometric approach. We start with a large square that has sides of length a plus b. Inside this square, we place four identical right triangles. The remaining space in the center forms a smaller square with side length c. By comparing the areas, we can derive the famous equation a squared plus b squared equals c squared.
Let's work through a practical example of the Pythagorean theorem. Imagine a ladder that is 5 meters long leaning against a wall. The base of the ladder is 3 meters away from the wall. We want to find how high up the wall the ladder reaches. Using the theorem, we set up the equation: 3 squared plus h squared equals 5 squared. This gives us 9 plus h squared equals 25. Solving for h, we get h squared equals 16, so h equals 4 meters.
The Pythagorean theorem has countless applications in our daily lives and various fields. It's used in navigation systems, construction, computer graphics, engineering, and physics. One important application is the distance formula in coordinate geometry, which directly uses the Pythagorean theorem to find the distance between two points. There are also famous Pythagorean triples like 3, 4, 5 and 5, 12, 13, where all three numbers are whole numbers that satisfy the theorem.