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The coordinates of the vertices of [tex]\(\triangle JKL\)[/tex] are [tex]\(J(1,4), K(6,4)\)[/tex], and [tex]\(L(1,1)\)[/tex].
The coordinates of the vertices of [tex]\(\triangle J^{\prime}K^{\prime}L^{\prime}\)[/tex] are [tex]\(J^{\prime}(0,-4), K^{\prime}(-5,-4)\)[/tex], and [tex]\(L^{\prime}(0,-1)\)[/tex].

What is the sequence of transformations that maps [tex]\(\triangle JKL\)[/tex] to [tex]\(\triangle J^{\prime}K^{\prime}L^{\prime}\)[/tex]?

A sequence of transformations that maps [tex]\(\triangle JKL\)[/tex] to [tex]\(\triangle J^{\prime}K^{\prime}L^{\prime}\)[/tex] is a rotation of [tex]\(180^{\circ}\)[/tex] about the origin, followed by a translation.

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GinnyAnswer
To determine the correct sequence of transformations that maps [tex]$\triangle J K L$[/tex] to [tex]$\triangle J^{\prime} K^{\prime} L^{\prime}$[/tex], let's proceed with the detailed steps:

1. Identify the transformation:
- Given [tex]$\triangle J K L$[/tex] with vertices [tex]$J(1, 4)$[/tex], [tex]$K(6, 4)$[/tex], [tex]$L(1, 1)$[/tex].
- Given [tex]$\triangle J^{\prime} K^{\prime} L^{\prime}$[/tex] with vertices [tex]$J^{\prime}(0, -4)$[/tex], [tex]$K^{\prime}(-5, -4)$[/tex], [tex]$L^{\prime}(0, -1)$[/tex].

2. Analyzing the points:
- Observe similarities and differences between corresponding points.
- Notice that a transformation should map [tex]$(1, 4)$[/tex], [tex]$(6, 4)$[/tex], [tex]$(1, 1)$[/tex] to [tex]$(0, -4)$[/tex], [tex]$(-5, -4)$[/tex], [tex]$(0, -1)$[/tex], respectively.

3. Understanding the 180° Rotation:
- When a point [tex]$(x, y)$[/tex] is rotated 180° about the origin, it maps to [tex]$(-x, -y)$[/tex].
- Apply the 180° rotation to each vertex of [tex]$\triangle J K L$[/tex]:
- [tex]$J(1, 4)$[/tex] becomes [tex]$(-1, -4)$[/tex],
- [tex]$K(6, 4)$[/tex] becomes [tex]$(-6, -4)$[/tex],
- [tex]$L(1, 1)$[/tex] becomes [tex]$(-1, -1)$[/tex].

4. Comparing results:
- After the 180° rotation, we obtain the coordinates:
- [tex]$J$[/tex] corresponds to [tex]$(-1, -4)$[/tex],
- [tex]$K$[/tex] corresponds to [tex]$(-6, -4)$[/tex],
- [tex]$L$[/tex] corresponds to [tex]$(-1, -1)$[/tex].

This transformation shows how the original triangle's vertices match perfectly with the vertices of [tex]$\triangle J^{\prime} K^{\prime} L^{\prime}$[/tex] once rotated by 180° about the origin. Hence, the only necessary transformation is a rotation.

Therefore, the sequence of transformations that maps [tex]$\triangle J K L$[/tex] to [tex]$\triangle J^{\prime} K^{\prime} L^{\prime}$[/tex] is a rotation of [tex]$180^{\circ}$[/tex] about the origin.

A sequence of transformations that maps [tex]$\triangle J K L$[/tex] to [tex]$\triangle J^{\prime} K^{\prime} L^{\prime}$[/tex] is:

a rotation of [tex]$180^{\circ}$[/tex] about the origin.

No further transformations like translation, reflection, or an additional [tex]$90^{\circ}$[/tex] rotation are needed.
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