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Sagot :
Sure, let's analyze this question together step-by-step. The question involves performing a combination of transformations on a trapezoid ABCD to obtain its image A"B"C"D". The transformations include:
1. Translation by the vector (4, 0).
2. Reflection across the line y = x.
To determine the original coordinates of the trapezoid ABCD, we'll reverse these transformations.
### Step 1: Reverse the Reflection
The reflection across the line y = x swaps the coordinates of each point. Therefore, if A"B"C"D" has the coordinates (a", b"), the coordinates of A'B'C'D' (after reflection but before the translation) would be (b", a").
### Step 2: Reverse the Translation
The translation rule \( T_{4,0} \) implies that \( x \)-coordinates were increased by 4. To reverse this, we subtract 4 from the \( x \)-coordinate. Therefore, if the coordinates of A'B'C'D' are (x', y'), the coordinates of the original trapezoid ABCD would be (x' - 4, y').
Now let's use these rules to determine the possible coordinates of the pre-image trapezoid ABCD.
Given options for coordinates of the final image \( A"B"C"D" \):
- (1, -1), which becomes (1 - 4, -1) = (-3, -1)
- (1, 1), which becomes (1 - 4, 1) = (-3, 1)
- (7, -5), which becomes (7 - 4, -5) = (3, -5)
- (7, 0), which becomes (7 - 4, 0) = (3, 0)
None of the pre-images derived above are listed in the options, so we must have misunderstood something.
### Possible Coordinates of Trapezoid ABCD
Let's look at the transformation results for the options backwards:
1. Option (-1, 0):
- Translate back: (-1, 0) -> (-1 + 4, 0) = (3, 0)
- Reflect: (3, 0) -> (0, 3)
2. Option (-1, -5):
- Translate back: (-1, -5) -> (-1 + 4, -5) = (3, -5)
- Reflect: (3, -5) -> (-5, 3)
3. Option (1, 1):
- Translate back: (1, 1) -> (1 + 4, 1) = (5, 1)
- Reflect: (5, 1) -> (1, 5)
4. Option (7, 0):
- Translate back: (7, 0) -> (7 - 4, 0) = (3, 0)
- Reflect: (3, 0) -> (0, 3)
5. Option (7, -5):
- Translate back: (7, -5) -> (7 - 4, -5) = (3, -5)
- Reflect: (3, -5) -> (-5, 3)
### Conclusions:
Based on the reverse transformations, the original pre-image coordinates of trapezoid ABCD that match the provided options must be:
- Option (-1, 0) matches the pre-image coordinates of (-3, -1) after reverse transformation.
- Option (7, 0) matches (3, 0) but translates to (0, 3)
- Option (7, -5) matches (3,-5) but translates to (-5,3)
So the correct options that can name the coordinates of vertices of the pre-image, trapezoid ABCD are:
- Option (-1, 0)
- Option (7, 0)
1. Translation by the vector (4, 0).
2. Reflection across the line y = x.
To determine the original coordinates of the trapezoid ABCD, we'll reverse these transformations.
### Step 1: Reverse the Reflection
The reflection across the line y = x swaps the coordinates of each point. Therefore, if A"B"C"D" has the coordinates (a", b"), the coordinates of A'B'C'D' (after reflection but before the translation) would be (b", a").
### Step 2: Reverse the Translation
The translation rule \( T_{4,0} \) implies that \( x \)-coordinates were increased by 4. To reverse this, we subtract 4 from the \( x \)-coordinate. Therefore, if the coordinates of A'B'C'D' are (x', y'), the coordinates of the original trapezoid ABCD would be (x' - 4, y').
Now let's use these rules to determine the possible coordinates of the pre-image trapezoid ABCD.
Given options for coordinates of the final image \( A"B"C"D" \):
- (1, -1), which becomes (1 - 4, -1) = (-3, -1)
- (1, 1), which becomes (1 - 4, 1) = (-3, 1)
- (7, -5), which becomes (7 - 4, -5) = (3, -5)
- (7, 0), which becomes (7 - 4, 0) = (3, 0)
None of the pre-images derived above are listed in the options, so we must have misunderstood something.
### Possible Coordinates of Trapezoid ABCD
Let's look at the transformation results for the options backwards:
1. Option (-1, 0):
- Translate back: (-1, 0) -> (-1 + 4, 0) = (3, 0)
- Reflect: (3, 0) -> (0, 3)
2. Option (-1, -5):
- Translate back: (-1, -5) -> (-1 + 4, -5) = (3, -5)
- Reflect: (3, -5) -> (-5, 3)
3. Option (1, 1):
- Translate back: (1, 1) -> (1 + 4, 1) = (5, 1)
- Reflect: (5, 1) -> (1, 5)
4. Option (7, 0):
- Translate back: (7, 0) -> (7 - 4, 0) = (3, 0)
- Reflect: (3, 0) -> (0, 3)
5. Option (7, -5):
- Translate back: (7, -5) -> (7 - 4, -5) = (3, -5)
- Reflect: (3, -5) -> (-5, 3)
### Conclusions:
Based on the reverse transformations, the original pre-image coordinates of trapezoid ABCD that match the provided options must be:
- Option (-1, 0) matches the pre-image coordinates of (-3, -1) after reverse transformation.
- Option (7, 0) matches (3, 0) but translates to (0, 3)
- Option (7, -5) matches (3,-5) but translates to (-5,3)
So the correct options that can name the coordinates of vertices of the pre-image, trapezoid ABCD are:
- Option (-1, 0)
- Option (7, 0)
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