Final answer:
Line f'g' is a dilation of line fg by a factor of 4 about the origin, and they both have the same slope of 1. Therefore, line f'g' is parallel to line fg, maintaining the slope but changing the specific points on the line. Option 3 is the correct answer.
Step-by-step explanation:
The student asks about the relationship between line fg, which passes through points (1, 1) and (3, 3), and line f'g' that is a dilation of line fg by a factor of 4 about the origin. To answer this, we first need to determine the slope of line fg. Using the two given points, the slope (m) can be calculated as (3-1)/(3-1), which equals 1. Therefore, line fg has a slope of 1 and passes through the origin since one of the points is (1,1).
When a line passes through the origin and is dilated about the origin, the angle of the line with respect to the horizontal axis remains unchanged, but the length of the line segments between points increases by the dilation factor. Therefore, the slope of line f'g' remains the same as that of line fg, which is 1 in this case. However, the specific points on line f'g' will be (1*4, 1*4) and (3*4, 3*4), which are (4, 4) and (12, 12). While the points change, the slope does not.
Hence, the correct statement about line f'g' in relation to line fg is that lines fg and f'g' are parallel to each other. This corresponds to option 3: 'Lines fg and f'g' are parallel to each other.'