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

Which ordered pairs name the coordinates of the 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]

Question

14-07-2024
Mathematics

Answered

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

Which ordered pairs name the coordinates of the 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]

Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2024-07-14T19:10:23-04:00

To find the coordinates of the vertices of the pre-image trapezoid ABCD after applying the given transformation, let's follow these steps:

1. Understand the Transformations:
- Translation [tex]$ T_{4,0} $[/tex] translates each point 4 units to the right.
- Reflection [tex]$ r_{y=x} $[/tex] reflects each point over the line [tex]$ y = x $[/tex].

2. Analyze the Given Points:
The coordinates provided are potential candidates for the pre-image vertices:
- [tex]$ (-1, 0) $[/tex]
- [tex]$ (-1, -5) $[/tex]
- [tex]$ (1, 1) $[/tex]
- [tex]$ (7, 0) $[/tex]
- [tex]$ (7, -5) $[/tex]

3. Apply the Translation:
Applying the translation [tex]$ T_{4,0} $[/tex] (4 units to the right):

- [tex]$ (-1, 0) $[/tex] becomes [tex]$ (3, 0) $[/tex]
- [tex]$ (-1, -5) $[/tex] becomes [tex]$ (3, -5) $[/tex]
- [tex]$ (1, 1) $[/tex] becomes [tex]$ (5, 1) $[/tex]
- [tex]$ (7, 0) $[/tex] becomes [tex]$ (11, 0) $[/tex]
- [tex]$ (7, -5) $[/tex] becomes [tex]$ (11, -5) $[/tex]

4. Apply the Reflection:
Applying the reflection [tex]$ r_{y=x} $[/tex] (swapping x and y coordinates):

- [tex]$ (3, 0) $[/tex] becomes [tex]$ (0, 3) $[/tex]
- [tex]$ (3, -5) $[/tex] becomes [tex]$ (-5, 3) $[/tex]
- [tex]$ (5, 1) $[/tex] becomes [tex]$ (1, 5) $[/tex]
- [tex]$ (11, 0) $[/tex] becomes [tex]$ (0, 11) $[/tex]
- [tex]$ (11, -5) $[/tex] becomes [tex]$ (-5, 11) $[/tex]

5. Determine the Pre-Image Coordinates:
To determine which points are actually part of the pre-image (i.e., vertices of trapezoid ABCD before any transformation), check the resulting coordinates to see if they match:
- [tex]$ (0, 3) $[/tex]
- [tex]$ (-5, 3) $[/tex]
- [tex]$ (1, 5) $[/tex]
- [tex]$ (0, 11) $[/tex]
- [tex]$ (-5, 11) $[/tex]

Since none of these coordinates appear to be among the given points, we conclude that after the transformations, these pre-image coordinates are not part of the original list of points mentioned.

Thus, upon careful consideration, it appears that none of the provided points are part of the vertices of the pre-image. The resulting list of pre-image vertices—taking into account the transformation steps—indicates no match with the given options.

The pre-image coordinates are indeterminable based on the options given after these sequences of transformations; hence, there are no correct choices among the provided coordinates.