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 y - axis, and then dilate that result with scale factor 2 centered at the origin.
• dilate figure a with scale factor 2 centered at the origin, and then reflect that result over the y - axis.
• dilate figure a with scale factor 2 centered at the origin, and then rotate that result clockwise 180° about the origin.
• dilate figure a with scale factor 2 centered at the origin, and then rotate that result clockwise 90° about the origin.
• none of these

(b) are figure a and figure b similar?

• yes - no

Explanation:

Part (a)

Step 1: Analyze Reflection and Dilation Order

First, consider reflecting Figure A over the \( y \)-axis. Then dilating with scale factor 2 centered at the origin. Alternatively, dilating first then reflecting. Let's check the shape and position. Figure A is a smaller version, Figure B is larger (scale factor 2) and on the right side (after reflection over \( y \)-axis or dilation then reflection).

• Reflect Figure A over \( y \)-axis: changes its horizontal position. Then dilate by 2: scales it.
• Dilate Figure A by 2 first: makes it larger, then reflect over \( y \)-axis: moves it to the right side. Both these sequences (reflect then dilate, or dilate then reflect) can map A to B because dilation preserves shape (similarity) and reflection is a rigid transformation. Rotation by \( 180^\circ \) or \( 90^\circ \) would change the orientation (Figure B has the same orientation as A after reflection/dilation, not rotation). So the first two options (reflect then dilate, dilate then reflect) work. Wait, let's re-examine:

Wait, Figure A is on the left of \( y \)-axis, Figure B is on the right. So reflecting over \( y \)-axis moves A to right side (mirror), then dilating by 2 (since B is larger) would make it match. Alternatively, dilating A (left) by 2 (making it larger left), then reflecting over \( y \)-axis moves it to right, which is B. Rotation by \( 180^\circ \) would flip both x and y, changing orientation (Figure B doesn't look rotated 180), rotation by \( 90^\circ \) would change shape orientation (B is same shape as A, just larger and mirrored). So the first two options: "Reflect Figure A over the \( y \)-axis, and then dilate that result with scale factor 2 centered at the origin" and "Dilate Figure A with scale factor 2 centered at the origin, and then reflect that result over the \( y \)-axis" are correct. The rotation options would change the orientation (Figure B has the same "triangle-like" shape with the right angle in the same corner, so rotation is not needed).

Step 2: Eliminate Rotation Options

Rotation by \( 180^\circ \) or \( 90^\circ \) would turn the figure, but Figure B has the same orientation as A (just mirrored and scaled). So the rotation options are incorrect. So the correct options are the first two: "Reflect Figure A over the \( y \)-axis, and then dilate that result with scale factor 2 centered at the origin" and "Dilate Figure A with scale factor 2 centered at the origin, and then reflect that result over the \( y \)-axis".

Part (b)

Step 1: Check Similarity

Similar figures have the same shape (corresponding angles equal, corresponding sides proportional). Since Figure B is a dilation (scale factor 2) of Figure A (with possible reflection, which is a rigid transformation), dilation preserves similarity (proportional sides, equal angles). So they are similar.

Answer:

(a):

• Reflect Figure A over the \( y \)-axis, and then dilate that result with scale factor 2 centered at the origin.
• Dilate Figure A with scale factor 2 centered at the origin, and then reflect that result over the \( y \)-axis.