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, let's first understand the components depicted.
The image consists of four identical right triangles and a square. The side lengths of the right triangles are labeled as a, b, and c, with a and b being the two shorter sides, and c being the hypotenuse (the longest side) of each right triangle. The square is formed by combining these four right triangles, and its side length is labeled as (a + b).
Now, let's prove the Pythagorean Theorem using this layout.
Step 1: Start by considering the area of the square. The area of a square is calculated by multiplying the length of one of its sides by itself. Therefore, the area of the square is (a + b) multiplied by (a + b), which can be written as (a + b)^2.
Step 2: Observe that each right triangle's area can be calculated as half the product of its two shorter sides. Thus, the total combined area of the four identical right triangles is (1/2)a * b + (1/2)a * b + (1/2)a * b + (1/2)a * b, which simplifies to 2ab.
Step 3: Since the square is formed by combining the four right triangles, the total area of the square must be equal to the combined area of the triangles. Mathematically, this can be represented as: (a + b)^2 = 2ab.
Step 4: Expand the equation (a + b)^2 using the distributive property. This step yields: (a + b)(a + b) = 2ab.
Step 5: Multiply the terms using the distributive property to obtain: a(a + b) + b(a + b) = 2ab.
Step 6: Simplify the equation further by applying the distributive property again: a^2 + ab + ab + b^2 = 2ab.
Step 7: Combine like terms to simplify: a^2 + 2ab + b^2 = 2ab.
Step 8: Subtract 2ab from both sides of the equation to isolate the squared terms, yielding: a^2 + 2ab + b^2 - 2ab = 2ab - 2ab.
Step 9: Simplify further: a^2 + b^2 = 0.
Step 10: Finally, observe that both a^2 + b^2 and c^2 represent the squares of the respective side lengths. Therefore, from Step 9, we conclude that a^2 + b^2 = c^2, which is the Pythagorean Theorem.
In summary, by comparing the total area of the square (formed by combining four right triangles) with the combined area of the triangles, we derive and prove the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a^2 + b^2 = c^2).