Answer :
To determine which sequences of transformations produce a congruent figure, we need to understand the types of transformations that preserve congruence. These transformations include translations, reflections, and rotations.
1. Translation: Moves every point of a figure the same distance in the same direction. (e.g., [tex]\( (x+a, y+b) \)[/tex]).
2. Reflection: Flips the figure over a line, which can change the sign of the coordinates. (e.g., [tex]\( (-x, y) \)[/tex] or [tex]\( (x, -y) \)[/tex]).
3. Rotation: Rotates the figure around a point by a certain angle, which maintains the sizes and shapes of figures.
Transformations that do not preserve congruence usually involve scaling, which changes the size of the figure.
Now we'll proceed to analyze each sequence provided:
1. [tex]\((x+2, 2y)\)[/tex]
- This transformation scales the y-coordinate by 2, which changes the size of the figure. Thus, it does not preserve congruence.
2. [tex]\((x+1, y-4)\)[/tex]
- This is a translation: moving every point 1 unit to the right and 4 units down. Translations preserve congruence.
3. [tex]\((x+2, y)\)[/tex]
- This is a translation: moving every point 2 units to the right. Translations preserve congruence.
4. [tex]\((-x, -2.5y)\)[/tex]
- This transformation involves a reflection over the y-axis (changing [tex]\(x\)[/tex] to [tex]\(-x\)[/tex]) and scaling of the y-coordinate by -2.5, which changes the size of the figure. Thus, it does not preserve congruence.
5. [tex]\((-x, y)\)[/tex]
- This transformation is a reflection over the y-axis (changing [tex]\(x\)[/tex] to [tex]\(-x\)[/tex]), which preserves congruence.
6. [tex]\((x-4, y+2)\)[/tex]
- This is a translation: moving every point 4 units to the left and 2 units up. Translations preserve congruence.
7. [tex]\((-x, 3y)\)[/tex]
- This transformation involves a reflection over the y-axis (changing [tex]\(x\)[/tex] to [tex]\(-x\)[/tex]) and scaling of the y-coordinate by 3, which changes the size of the figure. Thus, it does not preserve congruence.
8. [tex]\((x-2, y)\)[/tex]
- This is a translation: moving every point 2 units to the left. Translations preserve congruence.
By reviewing each transformation, we can conclude that the ones which maintain congruence (do not alter the shape or size) are:
- [tex]\((x+1, y-4)\)[/tex]
- [tex]\((x+2, y)\)[/tex]
- [tex]\((-x, y)\)[/tex]
- [tex]\((x-4, y+2)\)[/tex]
- [tex]\((x-2, y)\)[/tex]
Thus, the sequences of transformations that produce a congruent figure are:
[tex]\[ (x+1, y-4) (x+2, y) (-x, y) (x-4, y+2) (x-2, y) \][/tex]
Note that these transformations include translations and simple reflections, which do not alter the size or shape of the figure.
1. Translation: Moves every point of a figure the same distance in the same direction. (e.g., [tex]\( (x+a, y+b) \)[/tex]).
2. Reflection: Flips the figure over a line, which can change the sign of the coordinates. (e.g., [tex]\( (-x, y) \)[/tex] or [tex]\( (x, -y) \)[/tex]).
3. Rotation: Rotates the figure around a point by a certain angle, which maintains the sizes and shapes of figures.
Transformations that do not preserve congruence usually involve scaling, which changes the size of the figure.
Now we'll proceed to analyze each sequence provided:
1. [tex]\((x+2, 2y)\)[/tex]
- This transformation scales the y-coordinate by 2, which changes the size of the figure. Thus, it does not preserve congruence.
2. [tex]\((x+1, y-4)\)[/tex]
- This is a translation: moving every point 1 unit to the right and 4 units down. Translations preserve congruence.
3. [tex]\((x+2, y)\)[/tex]
- This is a translation: moving every point 2 units to the right. Translations preserve congruence.
4. [tex]\((-x, -2.5y)\)[/tex]
- This transformation involves a reflection over the y-axis (changing [tex]\(x\)[/tex] to [tex]\(-x\)[/tex]) and scaling of the y-coordinate by -2.5, which changes the size of the figure. Thus, it does not preserve congruence.
5. [tex]\((-x, y)\)[/tex]
- This transformation is a reflection over the y-axis (changing [tex]\(x\)[/tex] to [tex]\(-x\)[/tex]), which preserves congruence.
6. [tex]\((x-4, y+2)\)[/tex]
- This is a translation: moving every point 4 units to the left and 2 units up. Translations preserve congruence.
7. [tex]\((-x, 3y)\)[/tex]
- This transformation involves a reflection over the y-axis (changing [tex]\(x\)[/tex] to [tex]\(-x\)[/tex]) and scaling of the y-coordinate by 3, which changes the size of the figure. Thus, it does not preserve congruence.
8. [tex]\((x-2, y)\)[/tex]
- This is a translation: moving every point 2 units to the left. Translations preserve congruence.
By reviewing each transformation, we can conclude that the ones which maintain congruence (do not alter the shape or size) are:
- [tex]\((x+1, y-4)\)[/tex]
- [tex]\((x+2, y)\)[/tex]
- [tex]\((-x, y)\)[/tex]
- [tex]\((x-4, y+2)\)[/tex]
- [tex]\((x-2, y)\)[/tex]
Thus, the sequences of transformations that produce a congruent figure are:
[tex]\[ (x+1, y-4) (x+2, y) (-x, y) (x-4, y+2) (x-2, y) \][/tex]
Note that these transformations include translations and simple reflections, which do not alter the size or shape of the figure.
