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.
In the image described, we have a square made up of four right-angled triangles with sides a, b, and c as described. Let's focus on one of the right-angled triangles that make up the square, for example, the one in the top left corner.
By the Pythagorean theorem, we know that the sum of the squares of the two shorter sides (a and b) is equal to the square of the hypotenuse (c). In this case, we have:
a^2 + b^2 = c^2
Now, let's look at the inner square formed within the larger square. We can see that each side of the inner square (labeled c) forms the hypotenuse of one of the four right-angled triangles that make up the larger square.
Considering the right-angled triangle in the top left corner again, we can see that the two shorter sides a and b of this triangle are actually the two unequal parts of the side of the inner square labeled c. Therefore, we can rewrite the equation above as:
(2a)^2 + (2b)^2 = c^2
This simplifies to:
4a^2 + 4b^2 = c^2
Finally, since the inner square is a square, all sides are of equal length. Therefore, we have:
c^2 = 4a^2 + 4b^2
This proves the Pythagorean theorem, as we have shown that in this specific case, the sum of the squares of the two shorter sides is equal to the square of the longest side. This relationship holds true for all four right-angled triangles that make up the square, confirming the Pythagorean theorem.