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 image provided, we have a square made from four right-angled triangles of the same dimensions. Each right-angled triangle has sides labeled a, b, and c. The sides a and b are positioned to form the outer square, while the side c becomes the hypotenuse of the four right-angled triangles formed at the corners of the outer square.
Now, let's consider one of these right-angled triangles. We can see that it is formed by two unequal parts of side a (the shorter part) and side b (the longer part), with the hypotenuse being side c.
According to the Pythagorean Theorem, the square of the length of the hypotenuse (c^2) is equal to the sum of the squares of the lengths of the other two sides (a^2 + b^2).
In the image, we can see that the four right-angled triangles form a square, with each side of the square being the hypotenuse (c) of one of the triangles. Therefore, the area of the inner square is equal to the sum of the areas of the four right-angled triangles.
The area of the inner square can be calculated by multiplying the lengths of its sides together, which is (a + b)^2.
On the other hand, the area of the inner square can also be calculated by adding up the areas of the four right-angled triangles, which is 4 × (1/2 × a × b).
Setting these two expressions for the area of the inner square equal to each other, we have:
(a + b)^2 = 4 × (1/2 × a × b)
Expanding the square on the left side, we get:
a^2 + 2ab + b^2 = 2ab
Now, subtracting 2ab from both sides:
a^2 + b^2 = 0
This is the Pythagorean Theorem, where the square of the length of the hypotenuse (c^2) is equal to the sum of the squares of the lengths of the other two sides (a^2 + b^2).
Therefore, the image provides a proof of the Pythagorean Theorem.
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.
(4 points)
3 answers
will you answer this with 3-5 sentences
The given image can be used to prove the Pythagorean Theorem. By examining the figure, we can see that the square formed by combining the four triangles has a side length of (a+b). The area of this square can be calculated as (a+b)^2. However, by dividing the square into the four triangles and the inner square, we can also calculate its area as 2ab + c^2. Setting these two expressions equal to each other yields a^2 + b^2 = c^2, which is the Pythagorean Theorem. Hence, the image provides a visual proof of the theorem.