The Pythagorean theorem is one of the most famous mathematical formulas. It states that in a right triangle with legs of length a and b, and hypotenuse of length c, the relationship a squared plus b squared equals c squared always holds true.
To prove the Pythagorean theorem, we start with a geometric construction. First, we take our right triangle with sides a and b. Then we create a large square with side length a plus b. Inside this large square, we arrange four copies of our original triangle in the corners, leaving a smaller square in the center.
Now we calculate the area of the large square in two different ways. Method one: the area is simply a plus b squared, which expands to a squared plus 2ab plus b squared. Method two: we add up the areas of the four triangles plus the inner square. Each triangle has area one half ab, so four triangles give us 2ab. The inner square has area c squared.
Now we complete the algebraic proof. Step one: we expand the area of the large square, a plus b squared, which gives us a squared plus 2ab plus b squared. Step two: we set this equal to our second method, 2ab plus c squared. Step three: we subtract 2ab from both sides, leaving us with c squared equals a squared plus b squared. Therefore, we have proven the Pythagorean theorem!
The Pythagorean theorem has countless applications in real life. It's used for distance calculations, in engineering and construction, for navigation and GPS systems, in computer graphics, and throughout physics and astronomy. For example, with a triangle having sides 3 and 4, we can calculate that the hypotenuse is 5, since 3 squared plus 4 squared equals 25, which is 5 squared. This fundamental relationship continues to be one of the most useful tools in mathematics and science.