Unfortunately, as an AI text-based model, I cannot see or interpret images. However, I can explain one of the common proofs of the Pythagorean Theorem without the use of an image.
One of the commonly used proofs of the Pythagorean Theorem is known as the "Proof by rearrangement." Here's how it goes:
Consider a right-angled triangle with sides of lengths a, b, and c (where c is the hypotenuse), and let's denote the square with side length a as A, the square with side length b as B, and the square with side length c as C.
1. Start with the triangle and squares arranged as follows: place square A adjacent to the shorter side of the triangle (a), place square B adjacent to the other shorter side of the triangle (b), and finally, place square C adjacent to the hypotenuse (c).
2. Now, cut the square C into two rectangular shapes, as shown in the image below. This leaves us with four shapes: squares A and B, and two identical rectangles.
3. Rearrange the four shapes to form a large square as shown in the image. This square will have side length (a + b).
4. Notice that the rearranged square is composed of four congruent right-angled triangles, the original triangle is one of them. The other three triangles are formed by the rectangular shape resulting from cutting square C in the previous step.
5. Each of these four triangles is also congruent to one another because they have the same angle measurements and share a side. Therefore, they must also have the same area.
6. The area of the large square (a + b)^2 is equal to the sum of the areas of the four triangles and square C. Mathematically, this can be written as (a + b)^2 = 4 * (1/2) * a * b + c^2.
7. Simplify the equation by multiplying 4 * (1/2) * a * b, which gives 2ab. We now have (a + b)^2 = 2ab + c^2.
8. Rearrange the equation to isolate the terms with a and b on one side, which gives us (a + b)^2 - 2ab = c^2.
9. Simplify the left side of the equation by expanding the square: (a + b)^2 - 2ab = a^2 + 2ab + b^2 - 2ab = a^2 + b^2.
10. We are left with a^2 + b^2 = c^2, which is the Pythagorean Theorem.
So, in summary, the proof by rearrangement shows that by rearranging the sides of the original triangle and squares, we can form a larger square, and by comparing the areas of the resulting shapes, we can derive the Pythagorean Theorem.
Explain a proof of the Pythagorean Theorem using the image.
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