explain a proof of the Pythagorean theorem using the image. the image shows four right triangles close together forming a square on the inside
To prove the Pythagorean theorem using the given image, we start by considering the four right triangles formed by the sides of the large square:
1. Label the side lengths of the square as a, b, and c, with c being the hypotenuse (the longest side) of the right triangle.
2. Each of the right triangles has a hypotenuse equal to c, as the sides of the square are all congruent.
3. By rotating the four triangles within the square, we can rearrange them to form a smaller square in the center.
4. The side length of this smaller square is equal to a + b, as it is the sum of the two shorter sides of the right triangles.
5. To calculate the area of the smaller square, we can square the side length: (a + b)^2.
6. The total area of this square can also be determined by adding the areas of the four right triangles. Each triangle has an area of 1/2 * a * b.
7. Therefore, the total area of the smaller square is equal to 4 * (1/2 * a * b) = 2ab.
8. Equating the two expressions for the area of the smaller square, we have (a + b)^2 = 2ab.
9. Expanding the left side of the equation, we get a^2 + 2ab + b^2 = 2ab.
10. Simplifying further, we can subtract 2ab from both sides, yielding a^2 + b^2 = 0.
11. This equation represents the 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).
Therefore, the proof using the given image demonstrates the validity of the Pythagorean theorem.
in 1 to 2 sentence's
The image shows four right triangles arranged to form a square, and by equating the area of this square with the sum of the areas of the right triangles, the Pythagorean theorem is proven.
To prove the Pythagorean theorem using the given image, we can start by examining the square formed by the four smaller right triangles. Let's label the sides of this square as a, b, and c, where a and b are the legs of the right triangles and c is the hypotenuse.
The image represents that the area of the square is equal to the sum of the areas of the four right triangles. The area of each triangle is given by (1/2) * base * height, where the base is either a or b, and the height is the remaining side.
Therefore, the area of the square = (a * a) + (b * b) + (a * b) + (a * b) = a^2 + b^2 + 2ab.
Now, let's analyze how the square can be rearranged into two similar triangles.
If we draw a diagonal from one corner to another inside the square, it divides the square into two congruent right triangles. Let's call the length of this diagonal as d.
By rearranging the square, we see that the area of the square can also be expressed as the sum of the areas of these two triangles. Each triangle has an area of (1/2) * c * d.
Thus, the area of the square = (1/2) * c * d + (1/2) * c * d = c * d.
Since we established earlier that the area of the square is a^2 + b^2 + 2ab, and it is also equal to c * d, we can equate these two expressions:
a^2 + b^2 + 2ab = c * d
Recall that the diagonal d within the square is the same as the hypotenuse c of the right triangles.
So, we can rewrite the equation as:
a^2 + b^2 + 2ab = c^2
This is the Pythagorean theorem in its algebraic form, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.