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.
In the image, we can see four right-angled triangles with sides of lengths a, b, and c. The sides of the inner square are also labeled a, b, and c.
To prove the Pythagorean Theorem using this image, we can look at the relationship between the sides of the outer square and the sides of the inner square.
We can see that each side of the inner square is equal to the hypotenuse of one of the right-angled triangles. Therefore, the sum of the squares of the shorter sides of each triangle (a^2) and the longer sides of each triangle (b^2) is equal to the square of the hypotenuse of each triangle (c^2).
Since there are four right-angled triangles in the outer square, the sum of the squares of the shorter sides of each triangle (4a^2) and the sum of the squares of the longer sides of each triangle (4b^2) is equal to the sum of the squares of the hypotenuses of each triangle (4c^2).
Therefore, we can write the equation:
4a^2 + 4b^2 = 4c^2
Simplifying this equation, 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 is equal to the sum of the squares of the other two sides.