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 and simple
In the image, we can see that the area of the outer square is equal to the sum of the areas of the four right-angled triangles and the inner square.
The area of the outer square is (a + b)^2.
The area of each right-angled triangle is 1/2 * a * b. Since there are four triangles, the total area of the four triangles is 4 * 1/2 * a * b = 2ab.
The area of the inner square is c^2.
Therefore, (a + b)^2 = 2ab + c^2.
Expanding (a + b)^2 gives us a^2 + 2ab + b^2.
Substituting this back into the equation, we get a^2 + 2ab + b^2 = 2ab + c^2.
Simplifying further, a^2 + b^2 = c^2.
This proves 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).