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Looking at this proof of the Pythagorean Theorem, which statement(s) would be true? CHECK TWO THAT APPLY.

(2 points)
Responses

The length of a is the same as the length of c - (a+b).
The length of a is the same as the length of c - (a+b).

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 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.

length of a + length of b = length of c
length of a + length of b = length of c

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.
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.

The yellow triangles have an area of a x b, which would = c

1 answer

To determine which statements are true regarding the proof of the Pythagorean Theorem, let's analyze each statement. The Pythagorean Theorem states that for a right triangle with legs a and b, and hypotenuse c, the equation \( a^2 + b^2 = c^2 \) holds.

  1. The length of a is the same as the length of c - (a+b).

    • This statement is incorrect. The lengths do not relate in this way.

  2. 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 a little convoluted but essentially true. If the areas of the triangles are the same, and the configuration supports that, it leads to the conclusion about the areas of the squares.

  3. length of a + length of b = length of c

    • This is incorrect. The lengths of the legs of a right triangle do not add up to the length of the hypotenuse; rather, it is the areas that relate to the squares of these lengths.

  4. The area of both large squares (that are constructed on the legs of the triangle) are equal; so the triangles with the square(s) must have a total area that is equal to each other.

    • This statement is true. If the areas of the squares on the legs (a and b) are equal, then the remaining areas (the triangles) must relate to allow for an equality in total areas, supporting the theorem.

  5. The yellow triangles have an area of a x b, which would = c.

    • This statement is incorrect. The area representation given does not accurately convey the relationship in the context of the theorem. The product of a and b does not equate to c; rather, the areas are represented differently.

Conclusion

The two true statements are:

  • 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.