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The rule [tex]r_{y=x} \circ T_{4,0}(x, y)[/tex] is applied to trapezoid ABCD to produce the final image [tex]A^{-} B^{\prime} C^{\prime} D^{\prime}[/tex].

Which ordered pairs name the coordinates of vertices of the pre-image, trapezoid ABCD? Select two options.

A. [tex](-1, 0)[/tex]
B. [tex](-1, -5)[/tex]
C. [tex](1, 1)[/tex]
D. [tex](7, 0)[/tex]
E. [tex](7, -5)[/tex]

Sagot :

GinnyAnswer
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]
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