To explain a proof of the Pythagorean Theorem using an image, let’s consider a visual representation of the theorem itself.
The Pythagorean Theorem states that in a right 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 lengths of the other two sides. This can be expressed mathematically as:
\[ c^2 = a^2 + b^2 \]
where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides.
Proof Using a Square Configuration:
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Construct a large square: Imagine a large square whose side length is \( a + b \). The area of this large square is \( (a + b)^2 \).
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Four triangles inside the square: Inside this large square, we can arrange four identical right triangles. Each triangle has legs of lengths \( a \) and \( b \), and the hypotenuse \( c \). Placing these triangles in a way that their hypotenuses form a smaller square in the center.
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Calculating areas: The area of the large square can be calculated in two ways:
- First, by its side length: \( (a + b)^2 = a^2 + 2ab + b^2 \).
- Second, by considering the area of the four right triangles and the smaller square in the center: The area of one triangle is \( \frac{1}{2}ab \), so for four triangles, the total area is \( 4 \times \frac{1}{2}ab = 2ab \). The area of the smaller square formed by the hypotenuses is \( c^2 \). Thus, the area can also be expressed as \( 2ab + c^2 \).
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Setting the two area expressions equal: Now we can set the two expressions for the area of the large square equal: \[ a^2 + 2ab + b^2 = 2ab + c^2 \]
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Simplifying the equation: By canceling \( 2ab \) from both sides of the equation, we are left with: \[ a^2 + b^2 = c^2 \]
This concludes the proof of the Pythagorean Theorem using the area of squares and triangles. The visual representation helps to intuitively illustrate how the areas relate, confirming the theorem's validity through geometric means.
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