The image shows a box with uppercase letters 'A' and 'B' marked on two of its sides. In each of the four corners of the box, there are right-angled triangles formed with one side of the box as the hypotenuse. Inside each triangle, lowercase letter 'C' is labeled.
To prove the Pythagorean theorem, we will focus on one of the triangles. Let's take the bottom-left triangle, which is similar to the other three.
By the Pythagorean theorem, we know that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this triangle, the hypotenuse is line AB, and the other two sides are AC and BC. Let's consider their lengths.
We can see that line AC, starting from point A and ending at point C, is also the width of the box. Similarly, line BC, starting from point B and ending at point C, is equivalent to the length of the box.
If we consider the box as a square with side length AC, we can determine its area by multiplying AC by AC, giving us AC^2.
Likewise, we can consider the box as a square with side length BC, resulting in an area of BC^2.
Adding these two areas together, we have AC^2 + BC^2.
Now, let's examine the hypotenuse length, which is AB. We can see that AB is equal to the length of one side of the box plus the width of the box. Thus, AB is equal to AC + BC.
Simplifying AB^2, we have (AC + BC)^2, which expands to AC^2 + 2(AC)(BC) + BC^2.
Comparing AB^2 to AC^2 + BC^2, we notice that the expressions are equivalent based on their expansion. This implies that AC^2 + BC^2 = AB^2.
Therefore, we have proven the Pythagorean theorem for the particular case of the bottom-left triangle by demonstrating that the squares of the triangle's two shorter sides add up to the square of the hypotenuse.
Explain a proof of the Pythagorean theorem using the text describing an image.
A box with an A and a B on each side, and four triangles in the corners with C inside them.
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Summarize into 5 sentences
The image shows a box with uppercase letters 'A' and 'B' marked on two sides, and four right-angled triangles in the corners. The lowercase letter 'C' is inside each triangle. We focus on the bottom-left triangle which represents a right-angled triangle. By the Pythagorean theorem, the sum of the squares of the two shorter sides (AC and BC) is equal to the square of the hypotenuse (AB). Using the dimensions of the box, we illustrate that the area of the box can be represented as AC^2 + BC^2, while the length of AB can be expressed as AC + BC. Expanding the terms, we find that AC^2 + BC^2 is equal to (AC + BC)^2, thereby proving the Pythagorean theorem for this particular triangle.
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