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.

• dilate figure a with scale factor \\(\frac{1}{2}\\) centered at the origin, and then rotate that result counterclockwise \\(180^\circ\\) about the origin.
• dilate figure a with scale factor \\(\frac{1}{2}\\) centered at the origin, and then rotate that result clockwise \\(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 left 4 units.
• dilate figure a with scale factor \\(\frac{1}{2}\\) centered at the origin, and then reflect that result over the y - axis.
• none of these

(b) are figure a and figure b similar?

• yes
• no

Explanation:

Part (a)

Step 1: Analyze Dilation

First, dilate Figure A with scale factor $\frac{1}{2}$. This reduces the size of Figure A by half.

Step 2: Analyze Transformations

• Option 1 (180° rotation): A 180° rotation would flip the figure both horizontally and vertically, but after dilation, this doesn't match Figure B's position.
• Option 2 (90° clockwise rotation): A 90° clockwise rotation changes the orientation, not matching Figure B.
• Option 3 (Translate left 4 units): After dilating by $\frac{1}{2}$, translating left 4 units aligns the figure with Figure B's horizontal position.
• Option 4 (Reflect over y - axis): Reflecting over the y - axis would mirror the figure, not matching Figure B.
• Option 5 (None): Since Option 3 works, this is incorrect.

Step 3: Confirm Correct Option

The correct sequence is dilate with scale factor $\frac{1}{2}$ centered at the origin, then translate left 4 units. Also, for the rotation and reflection options, they don't give the correct orientation. The 180° rotation also doesn't work. The dilation - translation sequence works. Also, let's check the reflection and rotation again. When we dilate Figure A (the larger trapezoid) by $\frac{1}{2}$, we get a smaller trapezoid. Then, if we reflect over the y - axis, does it match? No. If we rotate 90° clockwise, the orientation is wrong. If we rotate 180°, the position is wrong. But if we translate left 4 units after dilation, it matches Figure B. Also, another way: dilate by $\frac{1}{2}$, then reflect over y - axis? Wait, no. Wait, maybe I made a mistake. Wait, Figure A is in the fourth quadrant (lower right), Figure B is in the second quadrant (upper left). Let's check the coordinates. Suppose a vertex of Figure A is at (4, - 2). After dilation by $\frac{1}{2}$, it's at (2, - 1). Then, if we reflect over the y - axis, it becomes (- 2, - 1), not matching. If we translate left 4 units, (2 - 4, - 1)=(- 2, - 1), still not. Wait, maybe the first option: dilate by $\frac{1}{2}$, then rotate 180°. Rotating 180° around the origin: (x,y)→(-x,-y). After dilation, (2, - 1)→(- 2,1), which is in the second quadrant, matching Figure B's position. Wait, I think I messed up earlier. Let's re - evaluate.

Let's take a vertex of Figure A. Let's assume Figure A has a vertex at (4, - 2) (approximate). Dilation by $\frac{1}{2}$: (2, - 1). Rotate 180°: (- 2,1), which is in the second quadrant, matching Figure B's position. Also, the shape: after dilation, the shape is similar, and 180° rotation aligns the orientation. Wait, maybe my initial analysis was wrong. Let's check the options again.

Wait, the first option: Dilate Figure A with scale factor $\frac{1}{2}$ centered at the origin, and then rotate that result counterclockwise 180° about the origin. A 180° counterclockwise rotation is the same as a 180° clockwise rotation, (x,y)→(-x,-y). After dilation, the coordinates are scaled by $\frac{1}{2}$, then rotated 180° gives (- $\frac{1}{2}$x, - $\frac{1}{2}$y) from the original (x,y) of Figure A.

The third option: Dilate, then translate left 4 units. Let's take a vertex of Figure A: say (6, - 3). Dilation by $\frac{1}{2}$: (3, - 1.5). Translate left 4 units: (3 - 4, - 1.5)=(- 1, - 1.5), which is in the third quadrant, not matching Figure B (which is in the second quadrant, upper left).

The fourth option: Dilate (6, - 3)→(3, - 1.5), reflect over y - axis: (- 3, - 1.5), third quadrant.

The second option: Dilate (6, - 3)→(3, - 1.5), rotate 90° clockwise: ( - 1.5, - 3), fourth quadrant (but scaled down, no).

The first option: Dilate (6, - 3)→(3, - 1.5), rotate…

Step 1: Definition of Similarity

Two figures are similar if one can be obtained from the other by a sequence of dilations and rigid transformations (rotations, reflections, translations). Since we can dilate Figure A to get a figure similar to Figure B (and then use a rigid transformation to map them exactly), they are similar.

Step 2: Confirm Similarity

Figure A and Figure B have the same shape (trapezoid), and we can perform a dilation (scale factor $\frac{1}{2}$) and a rigid transformation (rotation, etc.) to map one to the other, so they are similar.

Answer:

(a) The correct option is: Dilate Figure A with scale factor $\frac{1}{2}$ centered at the origin, and then rotate that result counterclockwise 180° about the origin. (Also, if the translation left 4 units is considered, but based on y - coordinate, the 180° rotation is correct. Wait, maybe the problem's figures have Figure A above? No, the original image shows Figure A below the x - axis and Figure B above. So the 180° rotation is correct. So the correct option is the first one: "Dilate Figure A with scale factor $\frac{1}{2}$ centered at the origin, and then rotate that result counterclockwise 180° about the origin" and also, maybe the third option is wrong. So the answer for (a) is the first option. For (b), the answer is Yes.

(a) \boxed{\text{Dilate Figure A with scale factor }\frac{1}{2}\text{ centered at the origin, and then rotate that result counterclockwise }180^\circ\text{ about the origin}} (and also, if there are multiple correct, but based on analysis, this is correct)
(b) \boxed{\text{Yes}}