Discover new perspectives and gain insights with IDNLearn.com. Ask your questions and receive reliable, detailed answers from our dedicated community of experts.
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)
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and see you next time for more reliable information.
