Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Connect with a community of experts ready to provide precise solutions to your questions on our user-friendly Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
To find the coordinates of the vertices of the pre-image trapezoid [tex]\(ABCD\)[/tex] given the final image vertices [tex]\(A'B'C'D'\)[/tex], we need to reverse the transformations applied to the image. The transformations involved are a reflection over the line [tex]\(y = x\)[/tex] followed by a translation by the vector [tex]\((4, 0)\)[/tex].
Let's go through the steps to reverse these transformations:
1. Reverse the Translation by (4, 0):
The translation moves each point [tex]\(4\)[/tex] units to the right. To reverse this, we need to move each point [tex]\(4\)[/tex] units to the left. Essentially, we will subtract [tex]\(4\)[/tex] from the x-coordinate of each point.
2. Reverse the Reflection over [tex]\(y = x\)[/tex]:
Reflecting a point over the line [tex]\(y = x\)[/tex] involves swapping its [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-coordinates. To reverse this reflection, we need to swap the coordinates back to their original positions.
We are given the following final image points to reverse engineer:
[tex]\[ (-1, 0), (-1, -5), (1, 1), (7, 0), (7, -5) \][/tex]
Let's find the corresponding pre-image points:
1. Point [tex]\((-1, 0)\)[/tex]:
- Reverse the Translation: [tex]\((-1 - 4, 0)\)[/tex] = [tex]\((-5, 0)\)[/tex]
- Reverse the Reflection: [tex]\((0, -5)\)[/tex]
2. Point [tex]\((-1, -5)\)[/tex]:
- Reverse the Translation: [tex]\((-1 - 4, -5)\)[/tex] = [tex]\((-5, -5)\)[/tex]
- Reverse the Reflection: [tex]\((-5, -5)\)[/tex]
3. Point [tex]\((1, 1)\)[/tex]:
- Reverse the Translation: [tex]\((1 - 4, 1)\)[/tex] = [tex]\((-3, 1)\)[/tex]
- Reverse the Reflection: [tex]\((1, -3)\)[/tex]
4. Point [tex]\((7, 0)\)[/tex]:
- Reverse the Translation: [tex]\((7 - 4, 0)\)[/tex] = [tex]\((3, 0)\)[/tex]
- Reverse the Reflection: [tex]\((0, 3)\)[/tex]
5. Point [tex]\((7, -5)\)[/tex]:
- Reverse the Translation: [tex]\((7 - 4, -5)\)[/tex] = [tex]\((3, -5)\)[/tex]
- Reverse the Reflection: [tex]\((-5, 3)\)[/tex]
The coordinates of the vertices of the pre-image trapezoid [tex]\(ABCD\)[/tex] after reversing the transformations are:
[tex]\[ (0, -5), (-5, -5), (1, -3), (0, 3), (-5, 3) \][/tex]
So, the correct ordered pairs for the pre-image of the vertices [tex]\(ABCD\)[/tex] are:
[tex]\[ (0, -5) \quad \text{and} \quad (-5, -5) \][/tex]
Therefore, we select the points:
[tex]\[ (-1,-5) \quad \text{and} \quad (7,-5) \][/tex]
Let's go through the steps to reverse these transformations:
1. Reverse the Translation by (4, 0):
The translation moves each point [tex]\(4\)[/tex] units to the right. To reverse this, we need to move each point [tex]\(4\)[/tex] units to the left. Essentially, we will subtract [tex]\(4\)[/tex] from the x-coordinate of each point.
2. Reverse the Reflection over [tex]\(y = x\)[/tex]:
Reflecting a point over the line [tex]\(y = x\)[/tex] involves swapping its [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-coordinates. To reverse this reflection, we need to swap the coordinates back to their original positions.
We are given the following final image points to reverse engineer:
[tex]\[ (-1, 0), (-1, -5), (1, 1), (7, 0), (7, -5) \][/tex]
Let's find the corresponding pre-image points:
1. Point [tex]\((-1, 0)\)[/tex]:
- Reverse the Translation: [tex]\((-1 - 4, 0)\)[/tex] = [tex]\((-5, 0)\)[/tex]
- Reverse the Reflection: [tex]\((0, -5)\)[/tex]
2. Point [tex]\((-1, -5)\)[/tex]:
- Reverse the Translation: [tex]\((-1 - 4, -5)\)[/tex] = [tex]\((-5, -5)\)[/tex]
- Reverse the Reflection: [tex]\((-5, -5)\)[/tex]
3. Point [tex]\((1, 1)\)[/tex]:
- Reverse the Translation: [tex]\((1 - 4, 1)\)[/tex] = [tex]\((-3, 1)\)[/tex]
- Reverse the Reflection: [tex]\((1, -3)\)[/tex]
4. Point [tex]\((7, 0)\)[/tex]:
- Reverse the Translation: [tex]\((7 - 4, 0)\)[/tex] = [tex]\((3, 0)\)[/tex]
- Reverse the Reflection: [tex]\((0, 3)\)[/tex]
5. Point [tex]\((7, -5)\)[/tex]:
- Reverse the Translation: [tex]\((7 - 4, -5)\)[/tex] = [tex]\((3, -5)\)[/tex]
- Reverse the Reflection: [tex]\((-5, 3)\)[/tex]
The coordinates of the vertices of the pre-image trapezoid [tex]\(ABCD\)[/tex] after reversing the transformations are:
[tex]\[ (0, -5), (-5, -5), (1, -3), (0, 3), (-5, 3) \][/tex]
So, the correct ordered pairs for the pre-image of the vertices [tex]\(ABCD\)[/tex] are:
[tex]\[ (0, -5) \quad \text{and} \quad (-5, -5) \][/tex]
Therefore, we select the points:
[tex]\[ (-1,-5) \quad \text{and} \quad (7,-5) \][/tex]
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.
