Question

select the correct answer. polygon abcd is reflected and dilated to give polygon pqrs. the coordinates of the preimage are (2, 2), (6, 8), (12, 8), and (16, 2). the coordinates of the image are (11, 15), (9, 12), (6, 12), and (4, 15). what is the scale factor of the dilation? a. 0.25 b. 0.33 c. 0.5 d. 0.75

Explanation:

Step1: Identify Corresponding Points

Find a pair of corresponding points from preimage (ABCD) and image (PQRS). Let's take point A(2, 2) and its image S(11, 15)? Wait, no, maybe better to check the vertical or horizontal distances. Wait, preimage points: A(2,2), B(6,8), C(12,8), D(16,2). Image points: S(11,15)? Wait, no, the image coordinates are (11,15), (9,12), (6,12), (4,15). Wait, maybe I mixed up. Wait, preimage ABCD: A(2,2), B(6,8), C(12,8), D(16,2). Image PQRS: let's match A to S? Wait, maybe the y-coordinates: preimage A(2,2), image S(11,15)? No, maybe the horizontal distance. Wait, preimage A(2,2) and image R(11,6)? Wait, no, the problem says preimage coordinates: (2,2), (6,8), (12,8), (16,2). Image coordinates: (11,15), (9,12), (6,12), (4,15). Wait, maybe I made a mistake. Wait, let's take the x-coordinates of A(2) and its image (let's see, image has (4,15), (6,12), (9,12), (11,15). Wait, maybe A(2,2) corresponds to (4,15)? No, that doesn't make sense. Wait, maybe the preimage is ABCD with A(2,2), B(6,8), C(12,8), D(16,2). The image is PQRS with S(11,15), R(9,12), Q(6,12), P(4,15). Wait, let's calculate the distance between A(2,2) and D(16,2): that's 16 - 2 = 14 units (horizontal). The distance between P(4,15) and S(11,15): 11 - 4 = 7 units. So scale factor is 7/14 = 0.5? Wait, no, 7 is the image length, 14 is preimage. Wait, 7/14 = 0.5? Wait, no, 7 is 14 * 0.5? Wait, 14 * 0.5 = 7. Yes. Alternatively, take the x-coordinate of A(2) and image P(4)? Wait, no, P is (4,15), A is (2,2). Wait, maybe the horizontal distance from A(2) to B(6) is 4 units. Image: from P(4) to R(9)? No, R is (9,12), Q is (6,12). Wait, preimage B(6,8) and image R(9,12)? No, maybe the vertical distance. Preimage B(6,8) has y=8, image R(9,12) has y=12. 12 - 8 = 4? No, 8 to 12 is +4, but 2 to 15? No, maybe I messed up. Wait, let's use the x-coordinates of A(2) and its image (let's see, image has (4,15), (6,12), (9,12), (11,15). Wait, A(2,2) and (4,15): x difference 2, y difference 13. B(6,8) and (6,12): x same, y difference 4. C(12,8) and (9,12): x difference -3, y difference 4. D(16,2) and (11,15): x difference -5, y difference 13. No, that's not dilation. Wait, maybe the preimage is ABCD with A(2,2), B(6,8), C(12,8), D(16,2). The image is PQRS with S(11,15), R(9,12), Q(6,12), P(4,15). Let's calculate the length of AB: distance between (2,2) and (6,8): $\sqrt{(6-2)^2 + (8-2)^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}$. Image length: between P(4,15) and R(9,12): $\sqrt{(9-4)^2 + (12-15)^2} = \sqrt{25 + 9} = \sqrt{34}$. No, that's not helpful. Wait, maybe the horizontal length of the preimage (ABCD) is from x=2 to x=16, so 14 units. The image (PQRS) is from x=4 to x=11, which is 7 units. So scale factor is 7/14 = 0.5? Wait, 14 * 0.5 = 7. Yes. Alternatively, take the x-coordinate of A(2) and image P(4): 4 - 2 = 2, but that's not. Wait, no, the preimage's A is (2,2), image's P is (4,15)? No, that's not. Wait, maybe I made a mistake in corresponding points. Let's look at the y-coordinates. Preimage A(2,2) and D(16,2) are on y=2. Image P(4,15) and S(11,15) are on y=15. The vertical distance between y=2 and y=15 is 13, but that's translation, not dilation. Wait, no, dilation is scaling, so the ratio of corresponding sides. Let's take the side AB: from (2,2) to (6,8). The vector is (4,6). The image side: from (4,15) to (9,12)? No, (4,15) to (9,12) is (5,-3). No, that's not. Wait, maybe the preimage is ABCD with A(2,2), B(6,8), C(12,8), D(16,2). The image is PQRS with S(11,15), R(9,12), Q(6,12), P(4,15). Let's calculate the length of AD: 16 - 2 = 14 (horizontal). Length of PS: 11 - 4 = 7 (horizontal). So scale factor is 7/14 = 0.5. Yes, that makes sense. So 7 is half of 14. So scale factor is 0.5.

Step2: Calculate Scale Factor

Scale factor = (Image Length) / (Preimage Length). For AD (preimage): 16 - 2 = 14. For PS (image): 11 - 4 = 7. So scale factor = 7/14 = 0.5.

Answer:

C. 0.5