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Sagot :
Let's analyze each option to determine if reversing the order of the transformations results in a different image of [tex]\(\triangle ABC\)[/tex].
### Option A:
Transformations:
1. Rotate [tex]\(90^\circ\)[/tex] counterclockwise about the origin.
2. Reflect across the [tex]\(y\)[/tex]-axis.
When you rotate [tex]\(\triangle ABC\)[/tex] [tex]\(90^\circ\)[/tex] counterclockwise, the points get new coordinates. Reflecting these new coordinates across the [tex]\(y\)[/tex]-axis changes their [tex]\(x\)[/tex]-values to their opposites. If you reverse the order:
1. Reflect across the [tex]\(y\)[/tex]-axis first.
2. Rotate [tex]\(90^\circ\)[/tex] counterclockwise.
This time, the points are reflected first, then the reflected points are rotated. The two different sequences will generally produce different images of [tex]\(\triangle ABC\)[/tex]. Hence, the transformations in Option A are not commutative.
### Option B:
Transformations:
1. Translate 5 units down.
2. Translate 4 units to the right.
Translating [tex]\(\triangle ABC\)[/tex] 5 units down and then 4 units to the right simply shifts the entire triangle. If you reverse the order:
1. Translate 4 units to the right.
2. Translate 5 units down.
The final position is the same regardless of the order because translation is commutative. Hence, the transformations in Option B are commutative.
### Option C:
Transformations:
1. Reflect across the [tex]\(x\)[/tex]-axis.
2. Reflect across the [tex]\(y\)[/tex]-axis.
First reflecting [tex]\(\triangle ABC\)[/tex] across the [tex]\(x\)[/tex]-axis changes the signs of the [tex]\(y\)[/tex]-coordinates. Then reflecting across the [tex]\(y\)[/tex]-axis changes the signs of the [tex]\(x\)[/tex]-coordinates. If you reverse the order:
1. Reflect across the [tex]\(y\)[/tex]-axis first.
2. Reflect across the [tex]\(x\)[/tex]-axis.
Each reflection changes the signs of the coordinates based on the order of the reflections. However, reflecting across both axes regardless of order results in a similar final position as both sequences negate both coordinates, leaving the same final image. Hence, the transformations in Option C are commutative.
### Option D:
Transformations:
1. Rotate [tex]\(180^\circ\)[/tex] clockwise about the origin.
2. Reflect across the [tex]\(y\)[/tex]-axis.
Rotating [tex]\(\triangle ABC\)[/tex] [tex]\(180^\circ\)[/tex] clockwise about the origin changes the coordinates to their respective opposites. Reflecting these points across the [tex]\(y\)[/tex]-axis changes their [tex]\(x\)[/tex]-coordinates. If you reverse the order:
1. Reflect across the [tex]\(y\)[/tex]-axis first.
2. Rotate [tex]\(180^\circ\)[/tex] clockwise.
Starting with a reflection across the [tex]\(y\)[/tex]-axis flips the [tex]\(x\)[/tex]-coordinates, and rotating [tex]\(180^\circ\)[/tex] will invert both coordinates once more but in a different order. This sequence will generally produce a different image of [tex]\(\triangle ABC\)[/tex]. Hence, the transformations in Option D are not commutative.
Based on this detailed analysis, the pair of transformations that will give a different image of [tex]\(\triangle ABC\)[/tex] if the order of the transformations is reversed is:
Answer: D. a rotation [tex]\(180^\circ\)[/tex] clockwise about the origin followed by a reflection across the [tex]\(y\)[/tex]-axis
### Option A:
Transformations:
1. Rotate [tex]\(90^\circ\)[/tex] counterclockwise about the origin.
2. Reflect across the [tex]\(y\)[/tex]-axis.
When you rotate [tex]\(\triangle ABC\)[/tex] [tex]\(90^\circ\)[/tex] counterclockwise, the points get new coordinates. Reflecting these new coordinates across the [tex]\(y\)[/tex]-axis changes their [tex]\(x\)[/tex]-values to their opposites. If you reverse the order:
1. Reflect across the [tex]\(y\)[/tex]-axis first.
2. Rotate [tex]\(90^\circ\)[/tex] counterclockwise.
This time, the points are reflected first, then the reflected points are rotated. The two different sequences will generally produce different images of [tex]\(\triangle ABC\)[/tex]. Hence, the transformations in Option A are not commutative.
### Option B:
Transformations:
1. Translate 5 units down.
2. Translate 4 units to the right.
Translating [tex]\(\triangle ABC\)[/tex] 5 units down and then 4 units to the right simply shifts the entire triangle. If you reverse the order:
1. Translate 4 units to the right.
2. Translate 5 units down.
The final position is the same regardless of the order because translation is commutative. Hence, the transformations in Option B are commutative.
### Option C:
Transformations:
1. Reflect across the [tex]\(x\)[/tex]-axis.
2. Reflect across the [tex]\(y\)[/tex]-axis.
First reflecting [tex]\(\triangle ABC\)[/tex] across the [tex]\(x\)[/tex]-axis changes the signs of the [tex]\(y\)[/tex]-coordinates. Then reflecting across the [tex]\(y\)[/tex]-axis changes the signs of the [tex]\(x\)[/tex]-coordinates. If you reverse the order:
1. Reflect across the [tex]\(y\)[/tex]-axis first.
2. Reflect across the [tex]\(x\)[/tex]-axis.
Each reflection changes the signs of the coordinates based on the order of the reflections. However, reflecting across both axes regardless of order results in a similar final position as both sequences negate both coordinates, leaving the same final image. Hence, the transformations in Option C are commutative.
### Option D:
Transformations:
1. Rotate [tex]\(180^\circ\)[/tex] clockwise about the origin.
2. Reflect across the [tex]\(y\)[/tex]-axis.
Rotating [tex]\(\triangle ABC\)[/tex] [tex]\(180^\circ\)[/tex] clockwise about the origin changes the coordinates to their respective opposites. Reflecting these points across the [tex]\(y\)[/tex]-axis changes their [tex]\(x\)[/tex]-coordinates. If you reverse the order:
1. Reflect across the [tex]\(y\)[/tex]-axis first.
2. Rotate [tex]\(180^\circ\)[/tex] clockwise.
Starting with a reflection across the [tex]\(y\)[/tex]-axis flips the [tex]\(x\)[/tex]-coordinates, and rotating [tex]\(180^\circ\)[/tex] will invert both coordinates once more but in a different order. This sequence will generally produce a different image of [tex]\(\triangle ABC\)[/tex]. Hence, the transformations in Option D are not commutative.
Based on this detailed analysis, the pair of transformations that will give a different image of [tex]\(\triangle ABC\)[/tex] if the order of the transformations is reversed is:
Answer: D. a rotation [tex]\(180^\circ\)[/tex] clockwise about the origin followed by a reflection across the [tex]\(y\)[/tex]-axis
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