Explain a proof of the Pythagorean Theorem using the image.

Answer 1

To prove the Pythagorean Theorem using the image, let us consider a right triangle ABC, where AB and BC are the two legs and AC is the hypotenuse.

Firstly, draw a square on each side of the right triangle, as shown in the image. The large square is drawn around the entire triangle ABC, and the two smaller squares are drawn on each leg of the triangle.

Next, observe that the area of the large square is equal to the sum of the areas of the two smaller squares. In other words, the area of the large square (ACDE) is equal to the area of the first small square (ABFG) plus the area of the second small square (BCHI).

Using the properties of squares, we can determine the areas of these squares. The area of the large square is given by the side length AC squared, denoted as AC^2. Similarly, the area of the first small square is AB^2, and the area of the second small square is BC^2.

Since the area of the large square is equal to the sum of the areas of the two smaller squares, we have AC^2 = AB^2 + BC^2. Voila! This equation is the Pythagorean Theorem.

Therefore, the image represents a proof of the Pythagorean Theorem by showing that the area of the square built on the hypotenuse equals the sum of the areas of the squares built on the two legs.