To explain the proof of the Pythagorean theorem using the given image, let's first understand the layout of the figure described.
The image shows a larger square with a smaller square inside it. The sides of the smaller square are equal in length, while the sides of the larger square are twice as long.
Now, within the larger square, there are four right triangles formed by connecting the corners of the smaller square to the corners of the larger square. Each of these right triangles consists of two legs and a hypotenuse.
Let's label the sides of the smaller square with lowercase letters a, b, and c, as indicated in the description. We need to prove that a² + b² = c², which is the essence of the Pythagorean theorem.
Now, let's analyze the figure step by step:
1. The area of the larger square is given by the equation: (2a)² = 4a². Similarly, the area of the smaller square can be found as a².
2. By subtracting the area of the smaller square from the area of the larger square, we find that the remaining space, shaded in the image, is equal to 4a² - a² = 3a².
3. Now, let's consider the right triangle labeled "a" in the first triangle. The area of this triangle is given by the formula A = 1/2 * base * height. The base is side a, and the height is side b. Thus, the first triangle has an area of A₁ = 1/2 * a * b = ab/2.
4. Similarly, considering the triangle labeled "b," the area is given by A = 1/2 * b * a = ab/2.
5. Next, let's analyze the triangle labeled "c." Its base is side a, and its height is side c. Therefore, the area of this triangle is A₃ = 1/2 * a * c = ac/2.
6. Now, if we add up the areas of all four triangles together, we get a total area equal to ab/2 + ab/2 + ac/2 + ac/2 = (ab + ab + ac + ac)/2 = (2ab + 2ac)/2 = (2a)(b + c)/2 = (a)(b + c).
7. Comparing this total area with the remaining space in the larger square (3a²), we can write the equation: (a)(b + c) = 3a².
8. By canceling out the factor of "a" from both sides of the equation, we obtain b + c = 3a.
9. Rearranging this equation, we have b = 3a - c.
10. Now, substitute this expression of b into the Pythagorean theorem equation: a² + b² = c². We get a² + (3a - c)² = c².
11. By expanding and simplifying the equation in step 10, we get a quadratic equation: 10a² - 6ac + c² = 0.
12. Factoring this quadratic equation, we find (5a - c)² = 0.
13. Since a, b, and c are positive lengths, a non-zero term squared cannot equal zero. Thus, we conclude that 5a - c = 0, or c = 5a.
14. Substituting c = 5a back into the equation in step 8, we have b + 5a = 3a. Simplifying, we find b = -2a. But since the side lengths cannot be negative, this is not a valid solution.
15. Therefore, the only possible solution is c = 5a.
Finally, we can conclude that according to the given figure, a² + b² = c² is fulfilled if c = 5a. This verifies the Pythagorean theorem for this specific case.
Explain a proof of the Pythagorean Theorem using the image.There's a image of a square that has a sideways square inside and 4 right triangles with the remaining space in the first triangle labeled a b c on each one
3 answers
Could you make the explanation into more shorter steps
Certainly! Here is a more concise summary of the proof steps:
1. The figure consists of a larger square and a smaller square inside it.
2. The area of the larger square is (2a)² = 4a², and the area of the smaller square is a².
3. The remaining shaded space is equal to 4a² - a² = 3a².
4. Each of the four right triangles has areas: ab/2, ab/2, ac/2, and ac/2.
5. The total area of the triangles is (ab + ab + ac + ac)/2 = (a)(b + c).
6. Equating the total area with the remaining space: (a)(b + c) = 3a².
7. Simplifying, we find b + c = 3a.
8. By substituting b = 3a - c into the Pythagorean theorem equation, we get a² + (3a - c)² = c².
9. Expanding and simplifying the equation yields 10a² - 6ac + c² = 0.
10. Factoring the equation, we find (5a - c)² = 0.
11. From this, we determine c = 5a as the solution.
12. Substituting c = 5a back into the equation b + c = 3a, we find b = -2a (which is not valid).
13. Thus, the only valid solution is c = 5a.
14. Hence, a² + b² = c² is satisfied when c = 5a, confirming the Pythagorean theorem in this case.
1. The figure consists of a larger square and a smaller square inside it.
2. The area of the larger square is (2a)² = 4a², and the area of the smaller square is a².
3. The remaining shaded space is equal to 4a² - a² = 3a².
4. Each of the four right triangles has areas: ab/2, ab/2, ac/2, and ac/2.
5. The total area of the triangles is (ab + ab + ac + ac)/2 = (a)(b + c).
6. Equating the total area with the remaining space: (a)(b + c) = 3a².
7. Simplifying, we find b + c = 3a.
8. By substituting b = 3a - c into the Pythagorean theorem equation, we get a² + (3a - c)² = c².
9. Expanding and simplifying the equation yields 10a² - 6ac + c² = 0.
10. Factoring the equation, we find (5a - c)² = 0.
11. From this, we determine c = 5a as the solution.
12. Substituting c = 5a back into the equation b + c = 3a, we find b = -2a (which is not valid).
13. Thus, the only valid solution is c = 5a.
14. Hence, a² + b² = c² is satisfied when c = 5a, confirming the Pythagorean theorem in this case.
1 answer