To determine which statements are true regarding the proof of the Pythagorean Theorem, we can analyze each response based on the context typically presented in such proofs.
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The yellow triangles have an area of a x b, which would = c.
- This statement seems incorrect. The area of each triangle would typically be \((1/2) \times \text{base} \times \text{height}\), which doesn't equal \(c\).
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The length of a is the same as the length of c - (a+b).
- This statement doesn't seem to be correct as well. It misrepresents the relationships among the sides based on the theorem.
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The four yellow triangles on the left all have the same area as the four yellow triangles on the right (a+b/2) - therefore, the two small green squares on the left have to have the same area as the large green square on the right.
- This statement is somewhat convoluted but reflects a common reasoning in Pythagorean proofs. If the areas are appropriately compared, this could be true, depending on how the triangles fit into the squares.
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The area of both large squares are equal; so the triangles with the square(s) must have a total area that is equal to each other.
- This statement is likely true, as the areas of the squares on the respective sides of the triangle reflect the Pythagorean theorem relationship \(a^2 + b^2 = c^2\).
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length of a + length of b = length of c.
- This statement is not true as per the Pythagorean theorem; it should state that \(c\) is the hypotenuse, which relates to \(a\) and \(b\) through the squared lengths.
Based on this analysis, the two true statements would be:
- The four yellow triangles on the left all have the same area as the four yellow triangles on the right (a+b/2) - therefore, the two small green squares on the left have to have the same area as the large green square on the right.
- The area of both large squares are equal; so the triangles with the square(s) must have a total area that is equal to each other.