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.
□ 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 rotate that result counterclockwise ( 90 ^ { circ } ) about the origin.
□ dilate figure a with scale factor ( \frac { 1 } { 2 } ) centered at the origin, and then translate that result to the right 5 units.
□ dilate figure a with scale factor ( \frac { 1 } { 2 } ) centered at the origin, and then reflect that result over the x - axis.
□ none of these
(b) are figure a and figure b similar?
yes no

Explanation:

Step1: Analyze the first option

• Rotate Figure A clockwise \(90^{\circ}\) about the origin: This changes the orientation. Then dilate with scale factor \(\frac{1}{2}\) centered at the origin: This changes the size. But the resulting figure after rotation and dilation does not match Figure B's position and orientation.

Step2: Analyze the second option

• Dilate Figure A with scale factor \(\frac{1}{2}\) centered at the origin (changes size). Then rotate counter - clockwise \(90^{\circ}\) about the origin (changes orientation). The combination of these two transformations maps Figure A onto Figure B.

Step3: Analyze the third option

• Dilate Figure A with scale factor \(\frac{1}{2}\) centered at the origin (changes size). Then translate to the right 5 units. Translation alone (after dilation) does not account for the orientation change needed to map Figure A to Figure B.

Step4: Analyze the fourth option

• Dilate Figure A with scale factor \(\frac{1}{2}\) centered at the origin (changes size). Then reflect over the x - axis (changes orientation). Reflection over the x - axis does not give the correct orientation to map Figure A to Figure B.

For part (b), since there is a sequence of similarity transformations (dilation and rotation) that can map Figure A to Figure B, the figures are similar.

Answer:

(a) Dilate Figure A with scale factor \(\frac{1}{2}\) centered at the origin, and then rotate that result counterclockwise \(90^{\circ}\) about the origin.
(b) Yes