Final answer:
After calculating the slopes of lines a and b, we find that line a has a slope of 1, while line b has a slope of 1.5. Since their slopes are neither equal nor negative reciprocals of one another, the lines are neither parallel nor perpendicular. So the correct answer is Option C.
Step-by-step explanation:
To determine whether lines a and b are parallel, perpendicular, or neither, we should find their slopes. The slope of a line is the ratio of the change in y to the change in x (rise over run).
For line a which passes through (1, 4) and (3, 6):
Slope of line a = (6 - 4) / (3 - 1) = 2 / 2 = 1.
For line b which passes through (-3, −6) and (−1, −3):
Slope of line b = (−3 - (−6)) / (−1 - (−3)) = 3 / 2 = 1.5.
Since line a and line b do not have the same slope, they are not parallel. Furthermore, for two lines to be perpendicular, the slope of one must be the negative reciprocal of the other (e.g., if one slope is a, the other should be -1/a). The slope of line a is 1, so the slope of line b would need to be -1 to be perpendicular, which it is not (it's 1.5). Hence, line a and line b are neither parallel nor perpendicular.