The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). To illustrate this proof using a square, let’s follow a geometric approach commonly associated with the theorem.
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Construct a Large Square: Begin by constructing a large square with a side length of (a + b). This square will contain four identical right triangles, each with legs of length a and b.
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Arrange the Triangles: Place the four triangles inside the large square. The triangles should be positioned such that their hypotenuses (c) face inward towards the center of the square. As a result, they form a smaller square in the middle of the larger square. The vertices of the smaller square are formed where the hypotenuses of the triangles meet.
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Calculate the Area of the Large Square: The area of the large square is calculated using the formula for the area of a square: \[ \text{Area}_{\text{large}} = (a + b)^2 = a^2 + 2ab + b^2 \]
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Calculate the Area of the Four Triangles and the Inner Square: Each right triangle has an area of: \[ \text{Area}{\text{triangle}} = \frac{1}{2}ab \] Since there are four triangles, the total area of the triangles is: \[ \text{Area}{\text{triangles}} = 4 \times \frac{1}{2}ab = 2ab \]
The area of the inner square (formed by the hypotenuses) is: \[ \text{Area}_{\text{inner}} = c^2 \]
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Set up the Equation: The area of the large square is equal to the total area of the four triangles plus the area of the inner square: \[ (a + b)^2 = 4 \left(\frac{1}{2}ab\right) + c^2 \]
This simplifies to: \[ a^2 + 2ab + b^2 = 2ab + c^2 \]
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Simplify and Conclude: By subtracting \(2ab\) from both sides, we find: \[ a^2 + b^2 = c^2 \]
This completes the proof of the Pythagorean Theorem. The arrangement of triangles and the areas within the square demonstrate that the sum of the squares of the two legs (a and b) equals the square of the hypotenuse (c).
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