To explain a proof of the Pythagorean Theorem using the description of your image, we'll rely on a geometric approach that involves squares and right triangles.
Pythagorean Theorem Overview
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.
Geometric Proof Explanation
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Constructing the Square: We start by drawing a large square. The side length of the square can be represented as \( a + b \), where \( a \) and \( b \) are the lengths of the two legs of the right triangle.
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Incorporating the Right Triangle: Inside this large square, we can place four identical right triangles, each having legs of length \( a \) and \( b \), positioned in such a way that the hypotenuse of each triangle is equal to \( c \). The arrangement forms a smaller square in the center, where the hypotenuses of the triangles meet.
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Calculating Area:
- Area of the Large Square: The area of the large square is \((a + b)^2\).
- Area of the Four Triangles: Since each triangle has an area of \(\frac{1}{2}ab\), the total area of the four triangles is \(4 \times \frac{1}{2}ab = 2ab\).
- Area of the Inner Square: The inner square formed at the center, where the hypotenuses meet, has a side length of \(c\). Therefore, its area is \(c^2\).
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Setting Up the Equation:
- From the total area, we have: \[ \text{Area of the large square} = \text{Area of the inner square} + \text{Area of the 4 triangles} \] This gives: \[ (a + b)^2 = c^2 + 2ab \]
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Expanding and Simplifying:
- Expanding the left side yields: \[ a^2 + 2ab + b^2 = c^2 + 2ab \]
- We can subtract \(2ab\) from each side: \[ a^2 + b^2 = c^2 \]
Conclusion
This geometric arrangement visually demonstrates the Pythagorean Theorem, proving that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse. The triangles are key to facilitating the understanding of how the areas relate within the square. Thus, the proof using squares, triangles, and areas leads us to the conclusion of the Pythagorean Theorem.