Use the image to answer the question.
A figure shows a square made from four right-angled triangles that all have the same dimensions. Each of the four right angled triangles have a height labeled a, a base labeled b, and a hypotenuse labeled c. Sides a and b are positioned so that the right angle creates the four outer corners of the outer square. Each vertex of the inner square divides each side of the outer square in two unequal parts labeled a and b, where a is the shorter part and b is the longer part. Each side of the inner square, labeled c, becomes the hypotenuse of the four right-angled triangles formed at the corners of the outer square. The four right-angled triangles are shaded.
Explain a proof of the Pythagorean Theorem using the image.
Short answer
A proof of the Pythagorean Theorem using the image can be explained as follows:
1. Let's assume that the length of side a (the shorter part) of the inner square is x.
2. Since each side of the inner square, labeled c, becomes the hypotenuse of the four right-angled triangles formed at the corners of the outer square, the hypotenuse of each of these triangles is also x.
3. Using the Pythagorean Theorem, we know that in a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b).
4. Therefore, in each of the four right-angled triangles formed at the corners of the outer square, we have c² = (x)² + (x)².
5. Simplifying the equation, we get c² = 2x².
6. Since there are four right-angled triangles, the sum of their areas is 4(c²/2) = 2c².
7. Now, let's consider the outer square. The area of the square is equal to the sum of the areas of the four right-angled triangles, which is 2c².
8. However, we can also calculate the area of the outer square as (a+b)².
9. Expanding (a+b)² = a² + 2ab + b², and since a and b are the sides of the square, they are equal in length, so we can simplify to 4a².
10. Equating the two expressions for the area of the outer square, we have 2c² = 4a².
11. Dividing both sides of the equation by 2, we get c² = 2a².
12. Comparing this result to step 5, we can conclude that 2a² = 2x².
13. Simplifying further, we get a² = x².
14. This equation proves the Pythagorean Theorem, as it states that the squares of the two smaller sides of a right-angled triangle are equal to the square of the hypotenuse.