Answer :
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.
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.
