Question

a dilation centered at the origin is applied to figure a. the result is figure b.
what is the scale factor of the dilation? write your answer as a whole number or a fraction in simplest form.

Explanation:

Step1: Identify corresponding sides

Let's assume the base of Figure B is \( b_B \) and the base of Figure A is \( b_A \). From the grid, suppose Figure B's base is 3 units and Figure A's base is 6 units (or other corresponding sides, but let's check the vertical sides too. Suppose Figure B's height is 2 units and Figure A's height is 4 units).

Step2: Calculate scale factor

The scale factor \( k \) from Figure A to Figure B (or vice versa, but since Figure B is smaller, it's a reduction). Scale factor is \( \frac{\text{Length of Figure B}}{\text{Length of Figure A}} \). If Figure B's side is 3 and Figure A's is 6, then \( k=\frac{3}{6}=\frac{1}{2} \). Wait, maybe I got the direction wrong. Wait, the problem says "a dilation centered at the origin is applied to Figure A. The result is Figure B". So Figure A is the original, Figure B is the image. So scale factor is \( \frac{\text{Image length}}{\text{Original length}} \). Let's check the bases. Let's count the grid squares. Suppose Figure A's base: from x=6 to x=12 (assuming), so length 6. Figure B's base: from x=2 to x=5, length 3. So scale factor \( \frac{3}{6}=\frac{1}{2} \). Or height: Figure A's height: from y=5 to y=9, length 4. Figure B's height: from y=2 to y=4, length 2. So \( \frac{2}{4}=\frac{1}{2} \). So scale factor is \( \frac{1}{2} \)? Wait, no, maybe Figure B is the original? Wait, the problem says "dilation centered at the origin is applied to Figure A. The result is Figure B". So Figure A is pre - image, Figure B is image. So scale factor \( k=\frac{\text{image size}}{\text{pre - image size}} \). So if Figure A is larger, Figure B is smaller, so \( k = \frac{1}{2} \)? Wait, maybe I mixed up. Wait, let's look at the grid again. Let's take the horizontal side (base) of Figure A: let's say from x = 6 to x = 12, so length 6. Figure B's base: from x = 2 to x = 5, length 3. So scale factor is \( \frac{3}{6}=\frac{1}{2} \). Or vertical side: Figure A's height: from y = 5 to y = 9, length 4. Figure B's height: from y = 2 to y = 4, length 2. So \( \frac{2}{4}=\frac{1}{2} \). So the scale factor is \( \frac{1}{2} \)? Wait, no, maybe Figure A is the image? Wait, the problem says "a dilation centered at the origin is applied to Figure A. The result is Figure B". So Figure A is transformed to Figure B. So Figure A is original, Figure B is image. So scale factor is \( \frac{\text{Figure B's side}}{\text{Figure A's side}} \). So if Figure A's side is 2 times Figure B's, then scale factor is \( \frac{1}{2} \). Wait, maybe I made a mistake. Let's count again. Let's assume the base of Figure A: let's see the coordinates. Suppose Figure A has a vertex at (6,5), (12,5), (6,9). So base length: 12 - 6 = 6, height: 9 - 5 = 4. Figure B has a vertex at (2,2), (5,2), (2,4). Base length: 5 - 2 = 3, height: 4 - 2 = 2. So scale factor: 3/6 = 1/2, 2/4 = 1/2. So the scale factor is \( \frac{1}{2} \). Wait, but maybe the other way? No, because dilation from A to B, so A is original, B is image. So scale factor is 1/2.

Answer:

\(\frac{1}{2}\)