Final answer:
On a coordinate plane, points that lie on the same line have equivalent ratios of change in y to change in x (slope). Evaluating the slope of the line passing through points (2, 1) and (6, 3) which is 0.5, the points that have the same slope and hence lie on this line are B (4, 2), D (8, 4), and E (12, 6).
Step-by-step explanation:
The question involves determining which points lie on a line in a coordinate plane that represents a set of equivalent ratios. The given line goes through points (2, 1) and (6, 3). To see which of the options given represent points on the same line, we need to check if the slope between the new points and one of the given points is consistent with the slope between the original points. The slope of the line can be calculated using the formula Δy/Δx, where Δ signifies a change in values.
First, we calculate the slope between the given points (2, 1) and (6, 3), which is (3 - 1) / (6 - 2) = 2 / 4 = 0.5. A point will lie on the same line if it forms the same slope with one of the points provided. Let's check each point:
- Point A (3, 2): The slope with (2, 1) is (2 - 1) / (3 - 2) = 1 / 1 = 1, which is not equal to 0.5, hence it does not lie on the line.
- Point B (4, 2): The slope with (2, 1) is (2 - 1) / (4 - 2) = 1 / 2 = 0.5, which is equal to 0.5, hence it lies on the line.
- Point C (4, 8): The slope with (2, 1) is (8 - 1) / (4 - 2) = 7 / 2, which is not equal to 0.5, hence it does not lie on the line.
- Point D (8, 4): The slope with (2, 1) is (4 - 1) / (8 - 2) = 3 / 6 = 0.5, which is equal to 0.5, hence it lies on the line.
- Point E (12, 6): The slope with (2, 1) is (6 - 1) / (12 - 2) = 5 / 10 = 0.5, which is equal to 0.5, hence it lies on the line.
Therefore, the points that are part of the set of equivalent ratios and lie on the same line are B (4, 2), D (8, 4), and E (12, 6).