Question

a sequence of transformations maps pentagon abcde onto congruent pentagon jklmn. select all the sequences of transformations that could have mapped pentagon abcde onto pentagon jklmn. a. a dilation by a scale factor of $\frac{2}{5}$ centered at the origin, then translate 1 unit left and 3 units up b. reflect over y - axis, then translate 1 unit right and 3 units up c. rotate 180°, then followed by a 90° clockwise rotation about the origin d. reflect over x - axis, then followed by a dilation by a scale factor of 3 centered at the origin e. a reflection across the line y = 2

Explanation:

Step1: Recall congruence - transformation rules

Congruent figures have the same shape and size. Dilations change the size of a figure.

Step2: Analyze Option A

A dilation by a scale factor other than 1 changes the size, so the figures won't be congruent. Option A is incorrect.

Step3: Analyze Option B

A reflection over the y - axis and then a translation is a rigid - motion. Rigid motions (reflections, translations, rotations) preserve congruence. Option B is correct.

Step4: Analyze Option C

Rotations are rigid motions. A 180° rotation followed by a 90° clock - wise rotation about the origin is a sequence of rigid motions that preserve congruence. Option C is correct.

Step5: Analyze Option D

A dilation by a scale factor of 3 changes the size of the figure, so the figures won't be congruent. Option D is incorrect.

Step6: Analyze Option E

A reflection across the line y = 2 is a rigid motion. Rigid motions preserve congruence. Option E is correct.

Answer:

B. reflect over y - axis, then translate 1 unit right and 3 units up
C. rotate 180°, then followed by a 90° clockwise rotation about the origin
E. a reflection across the line y = 2