Question

triangle abc was dilated and translated to form similar triangle abc. what is the scale factor of the dilation? \\(\frac{1}{5}\\) \\(\frac{2}{5}\\) \\(\frac{5}{2}\\) \\(\frac{5}{1}\\)

Explanation:

Step1: Identify side lengths of ABC

From the graph, triangle ABC has a horizontal side (AB) of length 1 (from x=0 to x=1) and vertical side (BC) of length 2 (from y=0 to y=2).

Step2: Identify side lengths of A'B'C'

Triangle A'B'C' has a horizontal side (A'B') of length 5 (from x=-4 to x=1) and vertical side (B'C') of length 5? Wait, no, let's check coordinates. Wait, maybe better to take corresponding sides. Wait, original ABC: let's find coordinates. A(0,2), B(1,2), C(1,0). So AB length: distance between A(0,2) and B(1,2) is 1. BC length: distance between B(1,2) and C(1,0) is 2. Now A'B'C': A'(-4,-2), B'(1,-2), C'(1,-7)? Wait, no, maybe I misread. Wait, the green triangle A'B'C': A'(-4,-2), B'(1,-2) (so AB length is 1 - (-4) = 5), and B'(1,-2) to C'(1,-7) (length 5? Wait, no, original BC was 2 (from y=2 to y=0, length 2). A'B'C' vertical side: from y=-2 to y=-7, length 5? Wait, no, maybe the vertical side of ABC is 2 (from y=0 to y=2? Wait, A(0,2), C(1,0): no, AC is the hypotenuse. Wait, maybe better to take AB and A'B'. AB is length 1 (from x=0 to x=1, y=2). A'B' is length 5 (from x=-4 to x=1, y=-2). So scale factor is length of A'B' / length of AB = 5/1? No, wait, maybe I got the direction wrong. Wait, dilation: if A'B'C' is the image, then scale factor is (length of image side)/(length of original side). Wait, original ABC: AB is 1 unit (from x=0 to x=1, same y). A'B' is 5 units (from x=-4 to x=1, same y). Wait, but maybe the vertical side: BC is 2 units (from y=0 to y=2? No, B(1,2), C(1,0): length 2 (from y=2 to y=0, so 2 units). B'C' is from y=-2 to y=-7? Wait, no, maybe the vertical side of A'B'C' is 5 units? Wait, no, let's recalculate. Wait, A(0,2), B(1,2), C(1,0). So AB: 1, BC: 2. A'(-4,-2), B'(1,-2), C'(1,-7). So A'B': 1 - (-4) = 5, B'C': -2 - (-7) = 5? No, that can't be. Wait, maybe I messed up the coordinates. Wait, the original triangle ABC: A(0,2), B(1,2), C(1,0). So AB is horizontal, length 1. BC is vertical, length 2 (from y=2 to y=0, so 2 units). The image triangle A'B'C': A'(-4,-2), B'(1,-2) (so A'B' length is 1 - (-4) = 5), and B'(1,-2) to C'(1,-7) (length 5? No, -2 to -7 is 5 units? Wait, original BC was 2 units (from y=0 to y=2? No, B(1,2) to C(1,0) is 2 units (downward). A'B'C' is downward from y=-2 to y=-7, length 5 units. So scale factor is 5/2? No, wait, maybe the vertical side of ABC is 2 (length 2) and vertical side of A'B'C' is 5? No, that doesn't match the options. Wait, the options are 1/5, 2/5, 5/2, 5/1. Wait, maybe I got the original and image reversed. If ABC is the image and A'B'C' is the original? No, the problem says "Triangle ABC was dilated and translated to form similar triangle A'B'C'". So ABC is the pre-image, A'B'C' is the image. So scale factor is (length of A'B') / (length of AB). AB length: 1 (from x=0 to x=1). A'B' length: 5 (from x=-4 to x=1). So 5/1? But that's 5, which is option D: 5/1. Wait, but let's check the vertical side. BC length: 2 (from y=0 to y=2? No, B(1,2) to C(1,0) is length 2 (down 2 units). B'C' length: from y=-2 to y=-7 is 5 units (down 5 units). So 5/2? Wait, no, maybe the horizontal side: AB is 1, A'B' is 5, so 5/1. But the options have 5/2. Wait, maybe I misread the coordinates. Let's re-express:

Original triangle ABC: points A(0,2), B(1,2), C(1,0). So AB: distance between (0,2) and (1,2) is 1. BC: distance between (1,2) and (1,0) is 2.

Image triangle A'B'C': points A'(-4,-2), B'(1,-2), C'(1,-7). So A'B': distance between (-4,-2) and (1,-2) is 5. B'C': distance between (1,-2) and (1,-7) is 5. Wait, no, that's 5, but original BC was 2. So scale factor would be 5/2? Wait, 5/2 is an option (option C: 5/2). Wait, maybe the vertical side of ABC is 2 (from y=0 to y=2? No, B(1,2) to C(1,0) is 2 units (y from 2 to 0, length 2). B'C' is from y=-2 to y=-7, length 5 (y from -2 to -7, length 5). So 5/2. Ah, that makes sense. So scale factor is (length of B'C') / (length of BC) = 5/2.

Step3: Calculate scale factor

Scale factor = (length of image side) / (length of pre - image side). Taking the vertical sides: BC (pre - image) has length 2, B'C' (image) has length 5. So scale factor = 5/2.

Answer:

C. $\frac{5}{2}$