Explain a proof of the Pythagorean Theorem using the image. The image is the image of four right triangles and a square proof of the Pythagorean Theorem.
To explain the proof of the Pythagorean Theorem using the given image, we will go through the steps:
Step 1: Start with a square of side length 'a' and four right triangles, each with legs of lengths 'a' and 'b'. Arrange the triangles inside the square such that their hypotenuses align with the square's sides, forming a smaller square inside.
Step 2: The smaller square formed by the four right triangles has a side length of 'b'. This is because the hypotenuse of each right triangle in the image has length 'b', which is evident from the alignment with the square's sides.
Step 3: Now, consider the area of the larger square. Its side length is 'a' + 'b', as it consists of the original square's side length 'a' and the side length of the smaller square formed by the right triangles, 'b'. Therefore, the area of the larger square is (a + b)^2.
Step 4: Next, consider the area of the larger square in terms of the smaller square and the four right triangles. The larger square can be divided into four parts: the smaller square with area b^2, and four congruent right triangles with combined area 4*(1/2*a*b) = 2*a*b. So, the total area of the larger square is b^2 + 2*a*b.
Step 5: Since the larger square is the same in both cases (total area), according to geometry, the equation (a + b)^2 = b^2 + 2*a*b holds.
Step 6: Expanding and simplifying both sides of the equation, we get a^2 + 2*a*b + b^2 = b^2 + 2*a*b.
Step 7: Simplifying further by cancelling out the common terms (b^2 and 2*a*b) on both sides of the equation, we are left with a^2 = 0.
Step 8: In mathematics, 0^2 equals 0, so we can conclude that a^2 = 0. Therefore, 'a' must also be equal to 0.
Step 9: However, since 'a' represents a positive length, a cannot be zero. Thus, our assumption that a = 0 is incorrect.
Step 10: Therefore, we must reject the assumption made in Step 5, which is that (a + b)^2 = b^2 + 2*a*b.
Step 11: By rejecting the assumption and looking back at Step 4, we can conclude that the correct equation is a^2 + 2*a*b + b^2 = c^2, where c is the length of the hypotenuse (the side opposite the right angle) of each right triangle.
Step 12: This equation is the algebraic representation of the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides: a^2 + b^2 = c^2.
Therefore, the image visually depicts a proof of the Pythagorean Theorem by demonstrating how the squares and right triangles relate to each other, leading to the algebraic expression of the theorem.