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Question
Which of the following sequence of transformations has been applied on 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 sequence of transformations has been applied on 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 A A 90∘ clockwise rotation about the point (1,1), Yes
Let’s check the transformations given in all the options and check if the image obtained is A′B′C′D′E′.
In option A: A 180∘ counterclockwise rotation around the origin, Yes
The above figure shows the counterclockwise rotation of pentagon ABCDE by 180∘ around the origin.
The above figure shows that image A′B′C′D′E′ is not matching with the image given in the question, which means this transformation is incorrect.
In option B: A 90∘ clockwise rotation about the point (1,1), Yes
The above figure shows the clockwise rotation of the pentagon ABCDE by 90∘ around the point (1,1).
The image A′B′C′D′E′ obtained in the above figure is the same as the image asked 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
Only 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 180∘ around (−3,0) and then translation of 2 units in the right, Yes
The above figure shows the counterclockwise rotation of ABCDE by 180∘ around (−3,0) to produce the dotted pentagon, and then its translation by 2 units toward the right direction to produce image A′B′C′D′E′. The image obtained in the above figure is not matching with the image given in the question. Thus, this sequence of transformation is incorrect.
➡Option B is correct.
Let’s check the transformations given in all the options and check if the image obtained is A′B′C′D′E′.
In option A: A 180∘ counterclockwise rotation around the origin, Yes
The above figure shows the counterclockwise rotation of pentagon ABCDE by 180∘ around the origin.
The above figure shows that image A′B′C′D′E′ is not matching with the image given in the question, which means this transformation is incorrect.
In option B: A 90∘ clockwise rotation about the point (1,1), Yes
The above figure shows the clockwise rotation of the pentagon ABCDE by 90∘ around the point (1,1).
The image A′B′C′D′E′ obtained in the above figure is the same as the image asked 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
Only 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 180∘ around (−3,0) and then translation of 2 units in the right, Yes
The above figure shows the counterclockwise rotation of ABCDE by 180∘ around (−3,0) to produce the dotted pentagon, and then its translation by 2 units toward the right direction to produce image A′B′C′D′E′. The image obtained in the above figure is not matching with the image given in the question. Thus, this sequence of transformation is incorrect.
➡Option B is correct.
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