Question

the parallelogram bcde is a dilation of the parallelogram bcde. 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 length of BC

Points B(-4,4) and C(4,4). Length BC: $|4 - (-4)| = 8$ (horizontal distance, y - coordinates same).

Step2: Find length of B'C'

Points B'(-1,1) and C'(1,1). Length B'C': $|1 - (-1)| = 2$ (horizontal distance, y - coordinates same).

Step3: Calculate scale factor

Scale factor = $\frac{\text{Length of image (B'C')}}{\text{Length of original (BC)}} = \frac{2}{8} = \frac{1}{4}$? Wait, no, wait. Wait, maybe I mixed up. Wait, B'C' is the image? Wait, no, wait the coordinates: Wait B is (-4,4), B' is (-1,1)? Wait no, looking at the graph: B is at (-4,4), B' is at (-1,1)? Wait no, the grid: Let's check coordinates again. Wait B: x=-4, y=4. B': x=-1, y=1? Wait no, the blue dots: B is at (-4,4), B' is at (-1,1)? Wait no, the B' is at (-1,1)? Wait no, looking at the graph, B' is at (-1,1)? Wait no, the coordinates: Let's see, the original BC: from x=-4 to x=4, so length 8 (since 4 - (-4) = 8). The B'C': from x=-1 to x=1, so length 2 (1 - (-1) = 2). Wait, but scale factor is image over original? Wait, but maybe I got original and image reversed. Wait the problem says B'C'D'E' is a dilation of BCDE. So BCDE is original, B'C'D'E' is image. So scale factor is (length of B'C') / (length of BC) = 2/8 = 1/4? Wait no, wait maybe I made a mistake in coordinates. Wait B is (-4,4), C is (4,4). So BC length: 4 - (-4) = 8 units (since horizontal line, y=4). B' is (-1,1), C' is (1,1). So B'C' length: 1 - (-1) = 2 units. So scale factor is 2/8 = 1/4? Wait but let's check another side. Original D is (0,-8), E is (-8,-8). So DE length: |-8 - 0|? No, E is (-8,-8), D is (0,-8). So DE length: 0 - (-8) = 8 units. D' is (0,-2), E' is (-2,-2). Wait E' is (-2,-2), D' is (0,-2). So D'E' length: 0 - (-2) = 2 units. So 2/8 = 1/4. Wait, but that seems small. Wait maybe I have original and image reversed. Wait maybe B'C'D'E' is the original? No, the problem says "B'C'D'E' is a dilation of BCDE", so BCDE is pre - image, B'C'D'E' is image. So scale factor is image length over pre - image length. So 2/8 = 1/4? Wait no, wait maybe I misread the coordinates. Wait B is at (-4,4), B' is at (-1,1)? Wait no, looking at the graph, B is at (-4,4), B' is at (-1,1)? Wait the y - coordinate of B is 4, B' is 1. The x - coordinate of B is -4, B' is -1. So the change in x: from -4 to -1 is +3, change in y: from 4 to 1 is -3. Wait, maybe the scale factor is 1/4? Wait no, wait 4 to 1 is multiplying by 1/4, -4 to -1 is multiplying by 1/4. Oh! Wait, the coordinates: B(-4,4), B'(-1,1). So to get from B to B', we multiply x by (-1)/(-4)=1/4, y by 1/4. Similarly, C(4,4) to C'(1,1): x:1/4, y:1/4. D(0,-8) to D'(0,-2): y: -2/-8 = 1/4. E(-8,-8) to E'(-2,-2): x: -2/-8 = 1/4, y: -2/-8 = 1/4. Ah, so the scale factor is 1/4? Wait no, wait -8 to -2: -2 / -8 = 1/4. 4 to 1: 1/4. So yes, scale factor is 1/4. Wait but let's confirm. Scale factor formula: for a dilation, the scale factor k is such that image coordinate = k original coordinate (if center is origin, which it seems here, since all coordinates are scaled by 1/4 from original to image? Wait no, original B(-4,4), image B'(-1,1): (-4)(1/4)= -1, 4*(1/4)=1. Yes! So the scale factor is 1/4. Wait, but I think I had a confusion earlier, but the key is to take the ratio of corresponding side lengths. So BC length is 8, B'C' length is 2, so 2/8 = 1/4.

Answer:

$\frac{1}{4}$