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.
To prove the Pythagorean Theorem using the given image, we can make use of the properties of the squares and the right-angled triangles formed.
1. Let's label the sides of the inner square as "x".
2. Each side of the square is divided into two parts, one of length "a" and the other of length "b". Therefore, the full length of each side of the outer square is "a + b".
3. Now, let's consider one of the right-angled triangles formed in the corners of the outer square. By the properties of triangles, we know that the sum of the lengths of the two shorter sides (the legs) is equal to the length of the hypotenuse. In this case, the shorter sides are "a" and "b" and the hypotenuse is "c".
4. By applying the Pythagorean Theorem to this triangle, we have: a^2 + b^2 = c^2.
5. Since we have four identical right-angled triangles in each corner of the outer square, we can multiply both sides of the equation by 4 to get: 4(a^2 + b^2) = 4c^2.
6. Now, let's look at the outer square. The length of each side is "a + b". Therefore, the area of the outer square is (a + b)^2 = (a + b)(a + b) = a^2 + b^2 + 2ab.
7. Since the outer square is composed of four identical right-angled triangles and an inner square, its area is also equal to the sum of their areas. The area of the outer square can be expressed as: 4c^2 + x^2.
8. We can equate the two expressions for the area: 4c^2 + x^2 = a^2 + b^2 + 2ab.
9. Rearranging the equation, we get: 4c^2 = a^2 + b^2 + 2ab - x^2.
10. Since the right-angled triangles and the inner square have the same dimensions, the areas of the four right-angled triangles (4c^2) and the area of the inner square (x^2) are equal: 4c^2 = x^2.
11. Substituting this into the previous equation, we have: x^2 = a^2 + b^2 + 2ab - x^2.
12. Simplifying, we get: 2x^2 = a^2 + b^2 + 2ab.
13. Dividing both sides of the equation by 2, we obtain: x^2 = (a^2 + b^2 + 2ab) / 2.
14. Finally, simplifying further, we have: x^2 = (a + b)^2 / 2.
15. Taking the square root of both sides, we get: x = (a + b) / √2.
16. Therefore, we have proven that the length of one side of the inner square (x) is equal to (a + b) / √2.
17. By rearranging the equation x = (a + b) / √2, we can express it as √2x = a + b.
18. Substituting this back into the Pythagorean Theorem equation we obtained earlier (a^2 + b^2 = c^2), we have: √2x^2 = c^2.
19. Taking the square root of both sides, we obtain: √2x = c.
20. Substituting this back into the previous expression (√2x = a + b), we have: √2x = a + b = c.
21. Therefore, we have proved the Pythagorean Theorem: a^2 + b^2 = c^2.