Question

the rectangle efgh is a dilation of the rectangle efgh. what is the scale factor of the dilation? simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

Explanation:

Step1: Find coordinates of corresponding points

For rectangle EFGH, let's take point G(0,8) and its image G'(0,2). Or take F(0,-4) and F'(0,-1) (wait, no, F' is (0,-1)? Wait, looking at the graph: F is at (0,-4), F' is at (0, -1)? Wait no, the grid: F is at (0,-4), F' is at (0, -1)? Wait no, the pink points: F' is at (0, -1)? Wait, no, the y-axis: F is at (0,-4), F' is at (0, -1)? Wait, no, let's check the length of FG. FG is from (0,-4) to (0,8), so length is 8 - (-4) = 12? Wait no, EFGH: F(0,-4), G(0,8), H(4,8), E(4,-4). So FG is vertical from (0,-4) to (0,8), length is 8 - (-4) = 12? Wait no, 8 - (-4) is 12? Wait, 8 - (-4) = 12? Wait, no, 8 - (-4) = 12? Wait, 8 + 4 = 12. Then F'G' is from (0,-1) to (0,2), length is 2 - (-1) = 3? Wait, no, F' is at (0, -1)? Wait, the pink points: F' is at (0, -1)? Wait, the graph: F' is at (0, -1)? Wait, the y-coordinate of F' is -1? Wait, no, the grid lines: each grid is 1 unit. So F is at (0, -4), F' is at (0, -1)? Wait, no, looking at the pink points: F' is at (0, -1)? Wait, no, the coordinates: F is (0, -4), F' is (0, -1)? Wait, no, G is (0,8), G' is (0,2). So the length of FG (vertical side) is 8 - (-4) = 12? Wait, no, 8 - (-4) is 12? Wait, 8 - (-4) = 12? Yes. Then F'G' is 2 - (-1)? Wait, no, G' is (0,2), F' is (0, -1)? Wait, no, F' is at (0, -1)? Wait, the pink rectangle: F' is (0, -1), E' is (1, -1), H' is (1,2), G' is (0,2). So F'G' is from (0,-1) to (0,2), length is 2 - (-1) = 3. Wait, but original FG is from (0,-4) to (0,8), length is 8 - (-4) = 12? Wait, that can't be. Wait, maybe I made a mistake. Let's check the horizontal side. EH is from (4,-4) to (4,8), length 12? No, E is (4,-4), H is (4,8), so length is 8 - (-4) = 12. Then E'H' is from (1,-1) to (1,2), length is 2 - (-1) = 3. So the scale factor is 3/12 = 1/4? Wait, no, wait G is (0,8), G' is (0,2). So the y-coordinate of G is 8, G' is 2. So the scale factor is 2/8 = 1/4. Ah, that's better. So take G(0,8) and G'(0,2). The scale factor is the ratio of the image length to the original length. So 2/8 = 1/4.

Step2: Calculate scale factor

Scale factor \( k = \frac{\text{Length of image segment}}{\text{Length of original segment}} \). For point G(0,8) and G'(0,2), the length of OG (from origin, but since it's vertical, the y-coordinate change) is 8 (from y=0 to y=8) and OG' is 2 (from y=0 to y=2). So scale factor \( k = \frac{2}{8} = \frac{1}{4} \).

Answer:

\(\frac{1}{4}\)