Welcome to our exploration of the Pythagorean theorem! This fundamental mathematical principle describes the relationship between the sides of a right triangle. The theorem states that a squared plus b squared equals c squared, where c is the hypotenuse.
Now let's see the visual proof! We can draw squares on each side of our right triangle. The red square has area a squared, the green square has area b squared, and the yellow square on the hypotenuse has area c squared. The theorem tells us that the area of the yellow square equals the combined areas of the red and green squares.
Let's work through a concrete example using the famous three-four-five triangle. We calculate three squared plus four squared, which equals nine plus sixteen, giving us twenty-five. And five squared also equals twenty-five. This confirms that three squared plus four squared equals five squared, proving the Pythagorean theorem for this triangle.
One of the most practical uses of the Pythagorean theorem is finding unknown side lengths. Here's an example: if we have a right triangle with sides of length six and eight, we can find the hypotenuse. Using the formula, c equals the square root of six squared plus eight squared, which equals the square root of one hundred, giving us ten.
To summarize what we've learned: The Pythagorean theorem is a fundamental relationship that applies only to right triangles. It states that a squared plus b squared equals c squared. This powerful tool helps us find unknown side lengths and has countless applications in mathematics, science, and everyday problem-solving.