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Question
Which of the following sequences of transformations has been applied to pentagon ABCDE to obtain A′B′C′D′E′ in the figure given below? Is the transformed image congruent to the preimage?
Which of the following sequences of transformations has been applied to pentagon ABCDE to obtain A′B′C′D′E′ in the figure given below? Is the transformed image congruent to the preimage?
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Solution
The correct option is D 90o clockwise rotation about the point (1,1), yes
Let’s check the transformation given in all options and check if the image obtained is A′B′C′D′E′.
In option A: 180o counterclockwise rotation around the origin, yes
The above figure shows the counterclockwise rotation of pentagon ABCDE by 180o around the origin.
We can see that image A′B′C′D′E′ does not match the image given in the question, which means this transformation is incorrect.
In option B: 90o clockwise rotation about the point (1,1), yes
The above figure shows the clockwise rotation of pentagon ABCDE by 90o around point (1,1).
The image A′B′C′D′E′ obtained in the above figure is the same as the image in the question, which means this transformation is correct.
Rotation is a rigid transformation and since rigid transformations always produce congruent images; therefore, A′B′C′D′E′ will be congruent to ABCDE.
Hence, option B is correct.
In option C: Translation by 2 units upward and then translation by 3 units toward the right, no
Just translation will not change the orientation of the image, and here, the orientation of ABCDE and A′B′C′D′E′ is quite different. Thus, this transformation is also incorrect.
In option D: Counterclockwise rotation of 180o around (−3,0) and then translation of 2 units to the right, yes
The above figure shows the counterclockwise rotation of ABCDE by 180o around (−3,0) to produce the dotted pentagon and then its translation by 2 units to the right to produce image A′B′C′D′E′. The image obtained in the above figure does not match the image given in the question. Thus, this sequence of transformations is incorrect.
Let’s check the transformation given in all options and check if the image obtained is A′B′C′D′E′.
In option A: 180o counterclockwise rotation around the origin, yes
The above figure shows the counterclockwise rotation of pentagon ABCDE by 180o around the origin.
We can see that image A′B′C′D′E′ does not match the image given in the question, which means this transformation is incorrect.
In option B: 90o clockwise rotation about the point (1,1), yes
The above figure shows the clockwise rotation of pentagon ABCDE by 90o around point (1,1).
The image A′B′C′D′E′ obtained in the above figure is the same as the image in the question, which means this transformation is correct.
Rotation is a rigid transformation and since rigid transformations always produce congruent images; therefore, A′B′C′D′E′ will be congruent to ABCDE.
Hence, option B is correct.
In option C: Translation by 2 units upward and then translation by 3 units toward the right, no
Just translation will not change the orientation of the image, and here, the orientation of ABCDE and A′B′C′D′E′ is quite different. Thus, this transformation is also incorrect.
In option D: Counterclockwise rotation of 180o around (−3,0) and then translation of 2 units to the right, yes
The above figure shows the counterclockwise rotation of ABCDE by 180o around (−3,0) to produce the dotted pentagon and then its translation by 2 units to the right to produce image A′B′C′D′E′. The image obtained in the above figure does not match the image given in the question. Thus, this sequence of transformations is incorrect.Suggest Corrections
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