To prove the Pythagorean Theorem using the given image, we can consider the areas of the squares and the triangles.
First, we know that the area of the larger, outer square is equal to the sum of the areas of the four smaller, shaded triangles and the smaller, inner square.
The area of the outer square is equal to the square of its side length, which is (a + b)^2.
The area of each triangle is equal to half the product of its base and height, which is 0.5ab.
The area of the inner square is equal to the square of its side length, which is c^2.
Therefore, we can write the equation: (a + b)^2 = 4(0.5ab) + c^2.
Simplifying this equation, we get: a^2 + 2ab + b^2 = 2ab + c^2.
Next, we can rearrange the equation to isolate c^2: a^2 + b^2 = c^2.
This equation matches the form of the Pythagorean Theorem, where a and b are the lengths of the legs of a right triangle, and c is the length of the hypotenuse.
Thus, the proof of the Pythagorean Theorem is demonstrated using the given image by comparing the areas of the squares and the triangles.
Question
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)
5 answers
make it smiple
Sure! Here's a simpler explanation:
In the image, we have a square made up of four triangles. Each triangle has a side labeled "a," another side labeled "b," and a diagonal labeled "c."
We can see that the outer square's side length is (a + b), and its area is (a + b)^2.
The smaller, shaded triangles have areas of 0.5ab each, so all four triangles combined have an area of 4(0.5ab) = 2ab.
The inner square has a side length of c and an area of c^2.
Using these measurements, we can set up the equation: (a + b)^2 = 2ab + c^2.
Simplifying further, we get: a^2 + 2ab + b^2 = 2ab + c^2.
By rearranging the equation, we isolate c^2: a^2 + b^2 = c^2.
This equation is the famous Pythagorean Theorem, which states that in a right triangle, the sum of the squares of the two shorter sides (a^2 + b^2) is equal to the square of the hypotenuse (c^2).
So, the image helps us understand the Pythagorean Theorem by comparing the areas of the squares and the triangles.
In the image, we have a square made up of four triangles. Each triangle has a side labeled "a," another side labeled "b," and a diagonal labeled "c."
We can see that the outer square's side length is (a + b), and its area is (a + b)^2.
The smaller, shaded triangles have areas of 0.5ab each, so all four triangles combined have an area of 4(0.5ab) = 2ab.
The inner square has a side length of c and an area of c^2.
Using these measurements, we can set up the equation: (a + b)^2 = 2ab + c^2.
Simplifying further, we get: a^2 + 2ab + b^2 = 2ab + c^2.
By rearranging the equation, we isolate c^2: a^2 + b^2 = c^2.
This equation is the famous Pythagorean Theorem, which states that in a right triangle, the sum of the squares of the two shorter sides (a^2 + b^2) is equal to the square of the hypotenuse (c^2).
So, the image helps us understand the Pythagorean Theorem by comparing the areas of the squares and the triangles.
make it shorter
Sure! Here's a shorter explanation:
In the image, a square is made up of four right-angled triangles with sides labeled a, b, and c. The outer square's side length is (a + b). By comparing the areas, we can prove the Pythagorean Theorem: a^2 + b^2 = c^2. This theorem states that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.
In the image, a square is made up of four right-angled triangles with sides labeled a, b, and c. The outer square's side length is (a + b). By comparing the areas, we can prove the Pythagorean Theorem: a^2 + b^2 = c^2. This theorem states that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.