The Pythagorean theorem is one of the most famous theorems in mathematics. It states that in any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. This relationship, expressed as a squared plus b squared equals c squared, forms the foundation for many geometric calculations and has practical applications in fields ranging from construction to computer graphics.
Now let's verify the Pythagorean theorem using a visual approach. We'll construct squares on each side of a 3-4-5 right 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. When we add the areas of the two smaller squares, 9 plus 16 equals 25, which exactly matches the area of the largest square. This visual proof demonstrates that the Pythagorean theorem holds true.
Now let's verify the Pythagorean theorem using algebraic methods and coordinate geometry. We place a right triangle on a coordinate system with vertices at A(0,0), B(3,0), and C(0,4). Using the distance formula, we calculate the length of each side. Side AB has length 3, side AC has length 4, and the hypotenuse BC has length 5. When we check our theorem, 3 squared plus 4 squared equals 9 plus 16, which equals 25, and this equals 5 squared. This algebraic verification confirms the Pythagorean theorem.
The dissection proof provides an elegant geometric demonstration of the Pythagorean theorem. We start with a large square that contains four identical right triangles arranged around a smaller inner square. The area of this configuration equals c squared. Next, we rearrange these same four triangles to form two separate rectangles, one with area a squared and another with area b squared. Since we're using the exact same triangles, the total area must remain constant. This proves that a squared plus b squared equals c squared, confirming the Pythagorean theorem through area conservation.
Now let's reinforce our understanding by verifying the Pythagorean theorem with multiple examples using different right triangles with integer sides. First, consider the 5-12-13 triangle: 5 squared plus 12 squared equals 25 plus 144, which equals 169, and this equals 13 squared. Next, the 8-15-17 triangle: 8 squared plus 15 squared equals 64 plus 225, which equals 289, and this equals 17 squared. Finally, the 7-24-25 triangle: 7 squared plus 24 squared equals 49 plus 576, which equals 625, and this equals 25 squared. All three examples confirm that the Pythagorean theorem holds true for these Pythagorean triples.