Use the image to answer the question. Explain a proof of the Pythagorean Theorem using the image
(hint: find the area of the colored triangles 2 ways. Then set them equal to each other)
C
b-a
C
a
b
The image displays a right triangle with sides labeled as a, b, and c. The right angle is formed at the bottom left vertex. The smaller triangle on the left has sides labeled as a and b-a, while the smaller triangle on the right has sides labeled as a and b. The larger triangle formed by combining the smaller triangles has side lengths a and b.
To prove the Pythagorean Theorem using this image, we can calculate the area of the larger triangle in two ways and equate them.
Method 1:
The area of the larger triangle is given by 1/2 * a * b.
Method 2:
We can calculate the areas of the two smaller triangles and add them up. The area of the smaller triangle on the left is 1/2 * (a) * (b-a) = 1/2 * a * b - 1/2 * a^2, and the area of the smaller triangle on the right is 1/2 * (a) * (b) = 1/2 * a * b.
Equating the areas from both methods:
1/2 * a * b = 1/2 * a * b - 1/2 * a^2 + 1/2 * a * b
Cancelling out the common terms on both sides, we have:
1/2 * a^2 = 1/2 * a^2
Since the equation is true, it serves as a proof of the Pythagorean Theorem, which states that in a right 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), or a^2 + b^2 = c^2.