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Reason |
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Consider a right-angled triangle with sides $a$, $b$, and hypotenuse $c$.
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Given.
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Construct a square with side length $a+b$.
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Geometric construction.
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Inside this square, arrange four copies of the triangle and a smaller square of side $c$.
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Visual arrangement.
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The area of the large square is $(a+b)^2$.
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Area of a square formula.
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The area of the four triangles is $4 \times \frac{1}{2}ab = 2ab$.
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Area of a triangle formula.
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The area of the inner square is $c^2$.
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Area of a square formula.
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The area of the large square is also the sum of the areas of the four triangles and the inner square:
$$ (a+b)^2 = 2ab + c^2 $$
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Area decomposition principle.
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Expanding the left side:
$$ a^2 + 2ab + b^2 = 2ab + c^2 $$
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Algebraic expansion.
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Subtract $2ab$ from both sides:
$$ a^2 + b^2 = c^2 $$
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Algebraic manipulation.
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