The Pythagorean theorem is one of the most famous theorems in mathematics. It states that in a right triangle with legs of lengths a and b, and hypotenuse of length c, we have a squared plus b squared equals c squared. Today we will prove this theorem using the area method, which involves constructing a geometric figure and calculating its area in two different ways.
Now we construct our proof figure. We start with a large square that has side length a plus b. Inside this large square, we place four identical copies of our right triangle. Each triangle is positioned so that its right angle vertex touches a corner of the large square. When we arrange the triangles this way, their hypotenuses form an inner square with side length c.
Now we calculate the area of our large square using the first method. Since the large square has side length a plus b, its area is simply a plus b squared. When we expand this expression, we get a squared plus 2ab plus b squared. This is our first way of calculating the total area.
Now we calculate the same area using the second method by adding up the areas of all the parts. The large square contains four identical right triangles, each with area one-half times a times b. So the total area of the four triangles is 2ab. The inner square has area c squared. Therefore, the total area is 2ab plus c squared.
Now we complete the proof by equating our two area calculations. Since both methods give the area of the same square, we have a plus b squared equals 2ab plus c squared. Expanding the left side gives us a squared plus 2ab plus b squared equals 2ab plus c squared. Subtracting 2ab from both sides, we get a squared plus b squared equals c squared. This is exactly the Pythagorean theorem, and our proof is complete!