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To determine which sequence of transformations produces a congruent figure, we need to review the types of transformations that preserve the size and shape of a figure. Congruent transformations include translations, reflections, and rotations but do not include scaling transformations (which change the size of the figure).
Let's analyze each option:
1. Option 1:
[tex]\[ \begin{array}{l} (-x, 3y) \\ (x-2, y) \end{array} \][/tex]
- First transformation: [tex]\((-x, 3y)\)[/tex]: This transformation involves reflecting over the [tex]\(y\)[/tex]-axis (congruent transformation) but also a scaling by 3 in the [tex]\(y\)[/tex]-direction. Scaling does not preserve congruence.
2. Option 2:
[tex]\((x+2, y)\)[/tex]
- This transformation is a translation of 2 units to the right along the [tex]\(x\)[/tex]-axis, which is a congruent transformation.
3. Option 3:
[tex]\((-x, -2.5y)\)[/tex]
- First transformation: reflecting over the [tex]\(y\)[/tex]-axis (congruent transformation).
- Second transformation: scaling by -2.5 along the [tex]\(y\)[/tex]-axis. Scaling (especially non-unitary and non-reflective) does not preserve congruence.
4. Option 4:
[tex]\[ \begin{array}{c} (x+2, 2y) \\ (x+1, y-4) \end{array} \][/tex]
- First transformation: [tex]\((x+2, 2y)\)[/tex]: This involves translating 2 units to the right (congruent transformation) but also scaling by 2 in the [tex]\(y\)[/tex]-direction (non-congruent).
- Second transformation: [tex]\((x+1, y-4)\)[/tex]: This is a translation 1 unit to the right and 4 units down (congruent transformation).
5. Option 5:
[tex]\[ \begin{array}{c} (-x, y) \\ (x-4, y+2) \end{array} \][/tex]
- First transformation: [tex]\((-x, y)\)[/tex]: This transformation is a reflection over the [tex]\(y\)[/tex]-axis, which is a congruent transformation.
- Second transformation: [tex]\((x-4, y+2)\)[/tex]: This is a translation 4 units to the left and 2 units up, which is also a congruent transformation.
Given this analysis, the sequence of transformations in option 5, [tex]\((-x, y)\)[/tex] followed by [tex]\((x-4, y+2)\)[/tex], are both congruent transformations (reflection and translation), making this option produce a congruent figure.
Therefore, the correct answer is:
[tex]$ (-x, y) , (x-4, y+2) $[/tex]
Let's analyze each option:
1. Option 1:
[tex]\[ \begin{array}{l} (-x, 3y) \\ (x-2, y) \end{array} \][/tex]
- First transformation: [tex]\((-x, 3y)\)[/tex]: This transformation involves reflecting over the [tex]\(y\)[/tex]-axis (congruent transformation) but also a scaling by 3 in the [tex]\(y\)[/tex]-direction. Scaling does not preserve congruence.
2. Option 2:
[tex]\((x+2, y)\)[/tex]
- This transformation is a translation of 2 units to the right along the [tex]\(x\)[/tex]-axis, which is a congruent transformation.
3. Option 3:
[tex]\((-x, -2.5y)\)[/tex]
- First transformation: reflecting over the [tex]\(y\)[/tex]-axis (congruent transformation).
- Second transformation: scaling by -2.5 along the [tex]\(y\)[/tex]-axis. Scaling (especially non-unitary and non-reflective) does not preserve congruence.
4. Option 4:
[tex]\[ \begin{array}{c} (x+2, 2y) \\ (x+1, y-4) \end{array} \][/tex]
- First transformation: [tex]\((x+2, 2y)\)[/tex]: This involves translating 2 units to the right (congruent transformation) but also scaling by 2 in the [tex]\(y\)[/tex]-direction (non-congruent).
- Second transformation: [tex]\((x+1, y-4)\)[/tex]: This is a translation 1 unit to the right and 4 units down (congruent transformation).
5. Option 5:
[tex]\[ \begin{array}{c} (-x, y) \\ (x-4, y+2) \end{array} \][/tex]
- First transformation: [tex]\((-x, y)\)[/tex]: This transformation is a reflection over the [tex]\(y\)[/tex]-axis, which is a congruent transformation.
- Second transformation: [tex]\((x-4, y+2)\)[/tex]: This is a translation 4 units to the left and 2 units up, which is also a congruent transformation.
Given this analysis, the sequence of transformations in option 5, [tex]\((-x, y)\)[/tex] followed by [tex]\((x-4, y+2)\)[/tex], are both congruent transformations (reflection and translation), making this option produce a congruent figure.
Therefore, the correct answer is:
[tex]$ (-x, y) , (x-4, y+2) $[/tex]
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