Question

answer the questions about figure a and figure b below.
(a) which sequence(s) of transformations will map figure a onto figure b exactly? choose all that apply.

• reflect figure a over the ( x )-axis, and then dilate that result with scale factor ( \frac{1}{2} ) centered at the origin.
• rotate figure a clockwise ( 90^circ ) about the origin, and then dilate that result with scale factor ( \frac{1}{2} ) centered at the origin.
• dilate figure a with scale factor ( \frac{1}{2} ) centered at the origin, and then reflect that result over the ( x )-axis.
• dilate figure a with scale factor ( \frac{1}{2} ) centered at the origin, and then rotate that result counterclockwise ( 180^circ ) about the origin.
• none of these

(b) are figure a and figure b similar?

• yes
• no

Explanation:

First, identify key vertices of Figure A: (2,2), (2,6), (4,6), (8,2)

Step1: Analyze Option 1 (Reflect then dilate)

Reflect over x-axis: (2,-2), (2,-6), (4,-6), (8,-2)
Dilate by $\frac{1}{2}$: $(1,-1)$, $(1,-3)$, $(2,-3)$, $(4,-1)$
This matches Figure B's vertices.

Step2: Analyze Option 2 (Rotate then dilate)

Rotate 90° clockwise: (2,-2), (6,-2), (6,-4), (2,-8)
Dilate by $\frac{1}{2}$: $(1,-1)$, $(3,-1)$, $(3,-2)$, $(1,-4)$
Does not match Figure B.

Step3: Analyze Option 3 (Dilate then reflect)

Dilate by $\frac{1}{2}$: $(1,1)$, $(1,3)$, $(2,3)$, $(4,1)$
Reflect over x-axis: $(1,-1)$, $(1,-3)$, $(2,-3)$, $(4,-1)$
This matches Figure B's vertices.

Step4: Analyze Option 4 (Dilate then rotate 180°)

Dilate by $\frac{1}{2}$: $(1,1)$, $(1,3)$, $(2,3)$, $(4,1)$
Rotate 180°: $(-1,-1)$, $(-1,-3)$, $(-2,-3)$, $(-4,-1)$
Does not match Figure B.

Step5: Address similarity question

Similar figures have proportional sides and congruent angles. Figure B is a scaled (scale factor $\frac{1}{2}$) and reflected version of Figure A, so all angles are congruent and sides are proportional.

Answer:

(a)

• Reflect Figure A over the x-axis, and then dilate that result with scale factor $\frac{1}{2}$ centered at the origin.
• Dilate Figure A with scale factor $\frac{1}{2}$ centered at the origin, and then reflect that result over the x-axis.

(b) Yes