I and III only. To be on the same line as (3, 5) and (4, 9), the slope between any given point and either (3, 5) or (4, 9) must be the same as the slope between (3, 5) and (4, 9). The (very) long way to do this problem would be to find the slope of (3, 5) and (4, 9). Using “change in y” divided by “change in x,” you get a slope of 4/1, or 4. Test the choice (2,1) with either (3, 5) or (4, 9) to see if the slope is thesame —for instance, the slope of the line segment between (2,1) and (3,5) is clearly 4/1, since the difference between the y-coordinates is 4 and the difference between the x-coordinates is 1. Since the slopes of these connecting line segments are the same, they are in fact parts of the same line.
It is possible to do this procedure for each choice. However, like most GRE problems, this problem has a “trick”: Using the original two points, notice that to get from (3, 5) to (4, 9), the x-coordinate goes up 1, while the y- coordinate goes up 4. Now just continue that pattern upward from (4, 9), adding 1 to the x and 4 to the y. You get (4 + 1, 9 + 4), or (5, 13). This point is not in the choices, and in fact you can now eliminate (5, 12) since the line passes above that point.
Keep going up on the line. (5 + 1, 13 + 4) is (6, 17), so this point is on the line. You can do the same trick going down. Start from (3, 5) and instead of adding 1 and 4, subtract 1 and 4. (3 - 1, 5 - 4)
is (2, 1), so this point is on the line.
Thus, (2, 1) and (6, 17) are on the line, and (5, 12) is not.
_________________