Whether two lines are parallel are perpendicular can be determined by comparing their slopes.
If the product of their slopes is -1, the lines are perpendicular. This property can also be called having negative reciprocals or having opposite reciprocals.
If the two lines have the same slope AND the y-intercepts are different, the lines are parallel. If both the slopes and the y-intercepts are equal, then the lines are classified as coincident. Essentially, you only have one line; you can also say you have one line right on top of the other.
To find the slope given two points, use the slope formula:
Slope = (y2 - y1) / (x1 - x2)
Substituting the first pair of coordinates into this equation results in:
(6-4)/((-2) -0) = 2/(-2) = -1
Doing the same with the second pair:
((-2) - 1)/(6-7) = (-3) / (-1) = 3
The slopes are not equal, so the lines are not parallel.
The product of the slopes is (-1) × 3 = -3. Since the product is not -1, the lines are not perpendicular.
So, the lines are neither parallel nor perpendicular.
NOTE that which point you pick for first and which you pick for second does not matter.
Reversing the order for the first pair:
(4-6) / (0 - (-2)) = (-2) / (2) = -1,
which is the same as we found before.
Reversing the order for the second pair,
(1 - (-2)) / (7-6) = 3/1 = 3,
which is also the same as we found before.
BUT make sure you don't switch the order of the y's but not the x's (nor switch the order of the x's, but not the y's). You will get the wrong answer if you do.
So, make sure whatever you pick for (x1,y1) and (x2,y2) are plugged in to the formula correctly. The y1 should be over x1 in the formula, and y2 should also be over x2.
James S.
10/12/23