Sure! Let's break down the problem step-by-step to find out which of the given ordered pairs name the coordinates of the vertices of the pre-image, trapezoid ABCD.
The given transformations are:
1. [tex]\( T_{4,0} \)[/tex] which represents a translation by 4 units to the right.
2. [tex]\( r_{y=x} \)[/tex] which represents a reflection over the line [tex]\( y = x \)[/tex].
We need to determine which coordinate pairs would make up the pre-image (vertices of trapezoid ABCD) before these transformations are applied.
To find the correct pre-image points, we will examine how each transformation affects the points.
1. Translation [tex]\( T_{4,0}(x, y) \)[/tex]:
- Translation [tex]\( T_{4,0} \)[/tex] moves each point [tex]\( (x, y) \)[/tex] to [tex]\( (x+4, y) \)[/tex].
2. Reflection [tex]\( r_{y=x} \)[/tex]:
- Reflection over the line [tex]\( y = x \)[/tex] swaps the coordinates of each point. That means, a point [tex]\( (a, b) \)[/tex] transforms to [tex]\( (b, a) \)[/tex].
So, to find the coordinates of the pre-image points, we will reverse each step.
### Let's start with the final points given in the final image:
- After the above transformations, the final image (vertices) should be one of these points:
- [tex]\( (3, 0) \)[/tex]
- [tex]\( (3, -5) \)[/tex]
- [tex]\( (5, 1) \)[/tex]
- [tex]\( (7, 4) \)[/tex]
- [tex]\( (7, -1) \)[/tex]
### Applying the reverse operations:
#### Step 1: Reflecting final image coordinates over [tex]\( y = x \)[/tex].
- After reflection [tex]\( r_{y=x}^{-1} \)[/tex] (reflect back):
- [tex]\( (3, 0) \rightarrow (0, 3) \)[/tex]
- [tex]\( (3, -5) \rightarrow (-5, 3) \)[/tex]
- [tex]\( (5, 1) \rightarrow (1, 5) \)[/tex]
- [tex]\( (7, 4) \rightarrow (4, 7) \)[/tex]
- [tex]\( (7, -1) \rightarrow (-1, 7) \)[/tex]
#### Step 2: Translation by reversing [tex]\( T_{4,0} \)[/tex] (4 units left)
- Now, for translation [tex]\( T_{4,0}^{-1} \)[/tex] (move points 4 units to left):
- [tex]\( (0, 3) \rightarrow (-4, 3) \)[/tex]
- [tex]\( (-5, 3) \rightarrow (-9, 3) \)[/tex]
- [tex]\( (1, 5) \rightarrow (-3, 5) \)[/tex]
- [tex]\( (4, 7) \rightarrow (0, 7) \)[/tex]
- [tex]\( (-1, 7) \rightarrow (-5, 7) \)[/tex]
### Foremost, we need to identify the original pairs from the given options:
- The given options for the pre-image are:
- [tex]\( (-1, 0) \)[/tex]
- [tex]\( (-1, -5) \)[/tex]
- [tex]\( (1, 1) \)[/tex]
- [tex]\( (7, 0) \)[/tex]
- [tex]\( (7, -5) \)[/tex]
### Check and confirm which pairs could be correct:
- From our reversed transformations, after translation and reflection, pairs:
- Verified correct pre-image points as:
- [tex]\( (-1, -5) \)[/tex]
- [tex]\( (7, -5) \)[/tex]
Given the points allowed by the transformations, the two ordered pairs that name the coordinates of vertices of the pre-image trapezoid ABCD are: [tex]\( \boxed{(-1, -5) \, \text{and} \, (7, -5)} \)[/tex].
