To solve this problem, we'll need to apply a sequence of transformations to the vertices of the pre-image, the trapezoid ABCD. The transformation rule given is [tex]\( r_{y=x} \circ T_{4,0}(x, y) \)[/tex], which means we will first translate the points by 4 units in the x-direction and then reflect the result over the line [tex]\(y = x\)[/tex].
Here are the steps we can follow:
1. Translation:
- The first transformation is [tex]\( T_{4,0} \)[/tex], which translates the points by 4 units in the x-direction.
- For each point [tex]\((x, y)\)[/tex], the new coordinates after translation will be [tex]\( (x + 4, y) \)[/tex].
2. Reflection:
- The second transformation is [tex]\( r_{y=x} \)[/tex], which reflects the points over the line [tex]\( y = x \)[/tex].
- For each point [tex]\((x, y)\)[/tex], the new coordinates after the reflection will be [tex]\((y, x)\)[/tex].
Let's apply these transformations to each of the given points:
1. Starting with the point [tex]\( (-1, 0) \)[/tex]:
- Translating: [tex]\( (-1 + 4, 0) = (3, 0) \)[/tex]
- Reflecting: [tex]\( (0, 3) \)[/tex]
2. Point [tex]\( (-1, -5) \)[/tex]:
- Translating: [tex]\( (-1 + 4, -5) = (3, -5) \)[/tex]
- Reflecting: [tex]\( (-5, 3) \)[/tex]
3. Point [tex]\( (1, 1) \)[/tex]:
- Translating: [tex]\( (1 + 4, 1) = (5, 1) \)[/tex]
- Reflecting: [tex]\( (1, 5) \)[/tex]
4. Point [tex]\( (7, 0) \)[/tex]:
- Translating: [tex]\( (7 + 4, 0) = (11, 0) \)[/tex]
- Reflecting: [tex]\( (0, 11) \)[/tex]
5. Point [tex]\( (7, -5) \)[/tex]:
- Translating: [tex]\( (7 + 4, -5) = (11, -5) \)[/tex]
- Reflecting: [tex]\( (-5, 11) \)[/tex]
From solving the problem, we establish that the correct coordinates of the pre-image vertices are two of these original points which matched as likely producing the final image after transformations:
- Given the results, the correct two options from the calculated pre-images are:
- [tex]\( (-1, 0) \)[/tex]
- [tex]\( (-1, -5) \)[/tex]
- [tex]\( (1, 1) \)[/tex]
So, the vertices of trapezoid ABCD as pre-images are [tex]\((-1, 0)\)[/tex], [tex]\((-1, -5)\)[/tex], and [tex]\((1, 1)\)[/tex].
