Question
These two images show steps in a proof of the Pythagorean theorem. Which of the following statements about the proof is false?
An image displaying two figures. Figure 1 has a big square that has two squares and two rectangles. One square with side a and another square with side b. The rectangle has length a and breadth b with diagonal c. Figure 2 has a big square that has a small square of side c and four triangles of side a, b and c.
A.
The sum of the areas of the triangles in step 1 equals the sum of the areas of the triangles in step 2.
B.
The proof shows that
.
C.
In step 1, the sum of the areas of the interior squares equals the sum of the areas of the four triangles.
D.
The area of the interior square in step 2 is equal to the combined area of the two interior squares in step 1
An image displaying two figures. Figure 1 has a big square that has two squares and two rectangles. One square with side a and another square with side b. The rectangle has length a and breadth b with diagonal c. Figure 2 has a big square that has a small square of side c and four triangles of side a, b and c.
A.
The sum of the areas of the triangles in step 1 equals the sum of the areas of the triangles in step 2.
B.
The proof shows that
.
C.
In step 1, the sum of the areas of the interior squares equals the sum of the areas of the four triangles.
D.
The area of the interior square in step 2 is equal to the combined area of the two interior squares in step 1
Answers
Answered by
GPT-4o mini
To determine which statement is false regarding the proof of the Pythagorean theorem based on the descriptions of the two figures, let's analyze each statement in light of the information provided about the two images.
A. **The sum of the areas of the triangles in step 1 equals the sum of the areas of the triangles in step 2.**
- This statement is likely true. Both steps involve triangles that correspond to the sides of the right triangle formed by the squares, and if the triangles are arranged correctly, their areas should be equal.
B. **The proof shows that \( a^2 + b^2 = c^2 \).**
- This statement is true. The essence of the Pythagorean theorem is to establish that the sum of the squares of the legs of a right triangle (sides \( a \) and \( b \)) equals the square of the hypotenuse (side \( c \)).
C. **In step 1, the sum of the areas of the interior squares equals the sum of the areas of the four triangles.**
- This statement is true as long as the areas of the squares are correctly represented. The area of the bigger square (which contains the two squares and four triangles) should equal the area of the interior squares plus the areas of the triangles.
D. **The area of the interior square in step 2 is equal to the combined area of the two interior squares in step 1.**
- This statement is true provided that the area represented by the interior square in step 2 (which is \( c^2 \)) is indeed equal to the combined areas of the squares in step 1 \( (a^2 + b^2) \).
Based on this analysis, all statements seem valid. To specifically identify which might be false, more context or specific visual details from the images would help, since the interpretation of the relationships between the areas is crucial. However, considering standard proofs of the Pythagorean theorem, it's plausible that:
**Confirm or question the areas stated in C, since it's a common point of confusion when arranging squares and triangles. If C states that the triangles made from those squares equal their combined area, this can occasionally be falsely interpreted.**
Ultimately, without further specific details or visuals, identifying a false statement outright is challenging, but C is a good candidate for scrutiny. If one must be chosen, C might inconsistently represent triangle arrangements.
A. **The sum of the areas of the triangles in step 1 equals the sum of the areas of the triangles in step 2.**
- This statement is likely true. Both steps involve triangles that correspond to the sides of the right triangle formed by the squares, and if the triangles are arranged correctly, their areas should be equal.
B. **The proof shows that \( a^2 + b^2 = c^2 \).**
- This statement is true. The essence of the Pythagorean theorem is to establish that the sum of the squares of the legs of a right triangle (sides \( a \) and \( b \)) equals the square of the hypotenuse (side \( c \)).
C. **In step 1, the sum of the areas of the interior squares equals the sum of the areas of the four triangles.**
- This statement is true as long as the areas of the squares are correctly represented. The area of the bigger square (which contains the two squares and four triangles) should equal the area of the interior squares plus the areas of the triangles.
D. **The area of the interior square in step 2 is equal to the combined area of the two interior squares in step 1.**
- This statement is true provided that the area represented by the interior square in step 2 (which is \( c^2 \)) is indeed equal to the combined areas of the squares in step 1 \( (a^2 + b^2) \).
Based on this analysis, all statements seem valid. To specifically identify which might be false, more context or specific visual details from the images would help, since the interpretation of the relationships between the areas is crucial. However, considering standard proofs of the Pythagorean theorem, it's plausible that:
**Confirm or question the areas stated in C, since it's a common point of confusion when arranging squares and triangles. If C states that the triangles made from those squares equal their combined area, this can occasionally be falsely interpreted.**
Ultimately, without further specific details or visuals, identifying a false statement outright is challenging, but C is a good candidate for scrutiny. If one must be chosen, C might inconsistently represent triangle arrangements.