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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.
I basically need help explaining based on the image.

Answer 1

Sure, I can help you explain a proof of the Pythagorean Theorem using the given image.

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In the given image, we have a square made up of four right-angled triangles. Each triangle has a right angle and sides labeled a, b, and c. We will use this image to prove the Pythagorean Theorem.

First, let's focus on the inner square. Since each side of the inner square is labeled as c, it becomes the hypotenuse of the four right-angled triangles formed at the corners of the outer square.

Now, let's look at the right-angled triangle on the top left corner of the outer square. This triangle has sides a, b, and c. By definition of a right-angled triangle, we know that a^2 + b^2 = c^2, as per the Pythagorean Theorem.

Similarly, using the right-angled triangles at the remaining three corners of the outer square, we can write the following equations:

- For the top right triangle: a^2 + b^2 = c^2
- For the bottom left triangle: a^2 + b^2 = c^2
- For the bottom right triangle: a^2 + b^2 = c^2

As we can see, all four triangles satisfy the Pythagorean Theorem equation.

Now, let's consider the outer square. Since the inner square is formed by joining the midpoints of the outer square's sides, we can see that the length of each side of the outer square is twice as long as the side length of the inner square.

Let's label the side length of the inner square as x, which means the side length of the outer square is 2x.

By using the Pythagorean Theorem on the outer square, we can write the equation:

(2x)^2 + (2x)^2 = (c + c)^2

Simplifying this equation, we get:

4x^2 + 4x^2 = 4c^2

Or, 8x^2 = 4c^2

Further simplifying, we get:

x^2 = c^2

This equation shows that the side length of the inner square squared (x^2) is equal to the hypotenuse squared (c^2).

Now, since the inner square represents the four right-angled triangles, we can conclude that the Pythagorean Theorem holds in this case as well:

a^2 + b^2 = c^2

Thus, the given image provides a visual proof of the Pythagorean Theorem, showing that it holds true in the context of the square made up of right-angled triangles.

Answer 2

Can you give me a smaller summary of this?

Answer 3

Certainly! Here's a shorter summary:

- The image shows a square made up of four right-angled triangles.
- The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
- The image provides a visual proof of the Pythagorean Theorem by showing that the equation holds true for each of the four right-angled triangles in the square.
- The side length of the inner square is defined as x, and the side length of the outer square is twice as long (2x).
- By applying the Pythagorean Theorem to the outer square, we can show that x^2 = c^2, confirming the validity of the theorem.
- In conclusion, the image provides a clear illustration of the Pythagorean Theorem in action.

Answer 4

Alright, Thanks Bot!

Answer 5

You're welcome! I'm glad I could help. If you have any more questions, feel free to ask.