explain a proof of the Pythagorean theorem using the image. the image shows four right triangles close together forming a square on the inside

Answer 1

To prove the Pythagorean theorem using the given image, we start by considering the four right triangles formed by the sides of the large square:

1. Label the side lengths of the square as a, b, and c, with c being the hypotenuse (the longest side) of the right triangle.

2. Each of the right triangles has a hypotenuse equal to c, as the sides of the square are all congruent.

3. By rotating the four triangles within the square, we can rearrange them to form a smaller square in the center.

4. The side length of this smaller square is equal to a + b, as it is the sum of the two shorter sides of the right triangles.

5. To calculate the area of the smaller square, we can square the side length: (a + b)^2.

6. The total area of this square can also be determined by adding the areas of the four right triangles. Each triangle has an area of 1/2 * a * b.

7. Therefore, the total area of the smaller square is equal to 4 * (1/2 * a * b) = 2ab.

8. Equating the two expressions for the area of the smaller square, we have (a + b)^2 = 2ab.

9. Expanding the left side of the equation, we get a^2 + 2ab + b^2 = 2ab.

10. Simplifying further, we can subtract 2ab from both sides, yielding a^2 + b^2 = 0.

11. This equation represents the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides (a^2 + b^2) is equal to the square of the hypotenuse (c^2).

Therefore, the proof using the given image demonstrates the validity of the Pythagorean theorem.

Answer 2

in 1 to 2 sentence's

Answer 3

The image shows four right triangles arranged to form a square, and by equating the area of this square with the sum of the areas of the right triangles, the Pythagorean theorem is proven.