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.

translate figure a to the right 8 units, and then dilate that result with scale factor \\( \frac { 1 } { 2 } \\) centered at the origin.

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.

dilate figure a with scale factor \\( \frac { 1 } { 2 } \\) centered at the origin, and then rotate that result clockwise \\( 90 ^ { \circ } \\) about the origin.

none of these
(b) are figure a and figure b similar?
yes no

Explanation:

Step1: Analyze the first option

Translate Figure A to the right 8 units, and then dilate that result with scale factor \(\frac{1}{2}\) centered at the origin.
After translation, the position is wrong for dilation to match Figure B.

Step2: Analyze the second option

Reflect Figure A over the \(x -\)axis, and then dilate that result with scale factor \(\frac{1}{2}\) centered at the origin.
Reflection over \(x -\)axis changes the \(y -\)coordinate sign. Then dilation with \(\frac{1}{2}\) doesn't map to Figure B.

Step3: Analyze the third option

Dilate Figure A with scale factor \(\frac{1}{2}\) centered at the origin, and then reflect that result over the \(x -\)axis.
Dilation first: coordinates are halved. Then reflection over \(x -\)axis (changes \(y -\)coordinate sign). This maps Figure A to Figure B.

Step4: Analyze the fourth option

Dilate Figure A with scale factor \(\frac{1}{2}\) centered at the origin, and then rotate that result clockwise \(90^{\circ}\) about the origin.
Rotation changes the orientation. After dilation and rotation, doesn't match Figure B.

For part (b):
Since there is a sequence of similarity transformations (dilation and reflection) that can map Figure A to Figure B, they are similar.

Answer:

a. Dilate Figure A with scale factor \(\frac{1}{2}\) centered at the origin, and then reflect that result over the \(x -\)axis.
b. Yes