Please prove the Pythagorean Theorem by combining graphics.
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The Pythagorean Theorem is one of the most famous theorems in mathematics. It states that in a right triangle, the square of the hypotenuse c equals the sum of the squares of the other two sides a and b. Today we'll prove this theorem using a beautiful visual method that rearranges geometric shapes to compare areas.
To prove the Pythagorean theorem, we construct a large square with side length a plus b. This square will serve as our canvas for the proof. We can place four identical right triangles inside this square in different arrangements. The total area of the large square is a plus b squared, which will remain constant regardless of how we arrange the triangles inside.
In our first arrangement, we place the four right triangles around a central square. The central square has side length c, which is the hypotenuse of our triangles, so its area is c squared. Each triangle has area one-half times a times b, and since we have four triangles, the total triangle area is 2ab. Therefore, the total area of our large square can be expressed as c squared plus 2ab.
Now we rearrange the same four triangles in a different way within the same large square. In this second arrangement, we create two separate squares instead of one central square. One square has side length a with area a squared, and the other has side length b with area b squared. The four triangles still have the same total area of 2ab. So our total area is now a squared plus b squared plus 2ab.
Now we can complete our proof. Since both arrangements use the same large square with area a plus b squared, and the same four triangles with total area 2ab, the remaining areas must be equal. From the first arrangement, we have c squared plus 2ab equals a plus b squared. From the second arrangement, we have a squared plus b squared plus 2ab equals a plus b squared. Therefore, c squared plus 2ab equals a squared plus b squared plus 2ab. Subtracting 2ab from both sides gives us c squared equals a squared plus b squared, which is exactly the Pythagorean theorem. This beautiful geometric proof demonstrates the theorem through pure area comparison.