Question

a series of transformations will be performed on a figure that is graphed on a coordinate - plane. which of the following series of transformations would produce a congruent figure? select the three correct answers. a a translation $\frac{1}{2}$ unit up followed by a reflection across b a reflection across the x - axis followed by a rotation of c a dilation by a scale factor of 2 followed by a translation 5 units d a dilation by a scale factor of - 1 followed by a reflection across the y - axis e a dilation by a scale factor of - 2 followed by a dilation by a scale factor of 2

Explanation:

Step1: Recall congruence - transformation rules

Rigid - motions (translations, rotations, reflections) preserve congruence. Dilations change the size of a figure and generally do not preserve congruence, except when the scale factor is 1 or - 1.

Step2: Analyze Option A

A translation $\frac{1}{2}$ unit up followed by a reflection across a line. Translations and reflections are rigid - motions. They preserve the size and shape of the figure, so the resulting figure is congruent.

Step3: Analyze Option B

A reflection across the x - axis followed by a rotation. Reflections and rotations are rigid - motions. They preserve the size and shape of the figure, so the resulting figure is congruent.

Step4: Analyze Option C

A dilation by a scale factor of 2 followed by a translation. A dilation with a scale factor of 2 changes the size of the figure, so the resulting figure is not congruent.

Step5: Analyze Option D

A dilation by a scale factor of - 1 followed by a reflection across the y - axis. A dilation with a scale factor of - 1 changes the orientation but not the size of the figure, and a reflection is a rigid - motion. The resulting figure is congruent.

Step6: Analyze Option E

A dilation by a scale factor of - 2 followed by a dilation by a scale factor of 2. The product of the scale factors $(-2)\times(2)= - 4$, which changes the size of the figure, so the resulting figure is not congruent.

Answer:

A. a translation $\frac{1}{2}$ unit up followed by a reflection across a line
B. a reflection across the x - axis followed by a rotation
D. a dilation by a scale factor of - 1 followed by a reflection across the y - axis