Given that the coordinates of the
points 𝐴, 𝐵, 𝐶, and 𝐷 are negative 15, eight; negative six, 10; negative eight,
negative seven; and negative six, negative 16, respectively, determine whether line
𝐴𝐵 and line 𝐶𝐷 are parallel, perpendicular, or neither.
Points 𝐴 and 𝐵 fall on the line
𝐴𝐵 and points 𝐶 and 𝐷 fall on the line 𝐶𝐷. To classify the lines, we have to
remember: parallel lines have the same slope and they do not intersect. Perpendicular lines have negative
reciprocal slopes and intersect at a 90-degree angle. And neither are lines that are not
parallel or perpendicular, lines that do intersect but do not form a right
angle. This means to consider whether or
not these lines are parallel or perpendicular, we need to know the slopes of these
In the general form, 𝑦 equals 𝑚𝑥
plus 𝑏, the 𝑚 represents the slope. And we can find the slope 𝑚 if we
have two points by saying 𝑚 equals 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥
one. In order to categorize these lines,
we need to find the slopes of line 𝐴𝐵 and line 𝐶𝐷. We can start with line 𝐴𝐵. Let point 𝐴 be 𝑥 one, 𝑦 one and
point 𝐵 be 𝑥 two, 𝑦 two. Then, the slope will be 10 minus
eight over negative six minus negative 15. 10 minus eight is two. Negative six minus negative 15 is
negative six plus 15, which is positive nine. So, we can say that the slope of
line 𝐴𝐵 is two-ninths.
We repeat this process for line
𝐶𝐷. Let 𝐶 be 𝑥 one, 𝑦 one and 𝐷 be
𝑥 two, 𝑦 two. And we’ll get 𝑚 equals negative 16
minus negative seven over negative six minus negative eight. Negative 16 minus negative seven is
negative 16 plus seven, which is negative nine. Negative six minus negative eight
is negative six plus eight which is two. The slope of line 𝐶𝐷 is then
negative nine over two.
If we compared these two slopes,
negative nine over two is the negative reciprocal of two over nine. And if you weren’t sure, you can
multiply them together. Reciprocals multiply together to
equal one and negative reciprocals multiply together to equal negative one. These two slopes are the negative
reciprocals of one another, making these lines perpendicular.