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.
The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
In the given image, we can see that each side of the inner square is the hypotenuse of one of the shaded triangles. Let's consider one of these triangles.
Using the labeling in the image, we can see that the square of the length of the hypotenuse c is equal to the sum of the squares of the lengths of the other two sides a and b. This can be represented mathematically as:
c^2 = a^2 + b^2
Now, let's consider the length of the side of the outer square. This side can be divided into two parts, with one part labeled a (the shorter part) and the other part labeled b (the longer part).
The length of the side of the outer square can be represented as:
a + b
Now, if we square the length of the side of the outer square, we get:
(a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2
Now, let's look at the area of the outer square. The area of a square is equal to the square of its side length. So, the area of the outer square can be represented as:
(a + b)^2
However, the area of the outer square can also be represented as the sum of the areas of the four shaded triangles plus the area of the inner square.
The area of each shaded triangle is equal to (1/2) * a * b, as each triangle is a right-angled triangle.
So, the total area of the four shaded triangles is:
4 * (1/2) * a * b = 2 * a * b
The area of the inner square is equal to the square of its side length, which is c.
So, the area of the outer square can be represented as:
2 * a * b + c^2
Since the area of the outer square is equal to the square of its side length, we can equate the two expressions for the area of the outer square:
(a + b)^2 = 2 * a * b + c^2
Expanding the left side, we get:
a^2 + 2ab + b^2 = 2 * a * b + c^2
Now, if we subtract 2ab from both sides, we get:
a^2 + b^2 = c^2
This is the Pythagorean Theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
Therefore, the given image can be used as a visual representation of the proof of the Pythagorean Theorem.