Video Transcript
Let 𝐿 be the line through the
points negative seven, negative seven and negative nine, six and 𝑀 the line
through one, one and 14, three. Which of the following is true
about the lines 𝐿 and 𝑀? Option (A) they are parallel,
option (B) they are perpendicular, or option (C) they are intersecting but not
perpendicular.
It might be worthwhile
beginning this question with a quick sketch of the two lines through the two
sets of points. When we do this, we can observe
that the two lines do in fact intersect. We can therefore say that these
two lines are not parallel, so we can eliminate option (A). Now, we can remember that two
lines are perpendicular if they intersect or meet at right angles. From the diagram, it does
appear that the two lines are at right angles. But it could be the case that
the two lines are nearly perpendicular and it’s not possible to distinguish this
from the diagram. Generally, it’s not a very good
idea to just use a sketch to determine if lines are parallel or
perpendicular. In fact, we should perform some
sort of calculation.
We can recall that if two
straight lines have slopes of 𝑚 sub one and 𝑚 sub two, then they are
perpendicular if 𝑚 sub two is equal to negative one over 𝑚 sub one. We’ll first need to calculate
the slopes of each of the lines 𝐿 and 𝑀. The slope of the line passing
through two points with coordinates 𝑥 sub zero, 𝑦 sub zero and 𝑥 sub one, 𝑦
sub one is calculated as the slope 𝑚 is equal to 𝑦 sub one minus 𝑦 sub zero
over 𝑥 sub one minus 𝑥 sub zero. For line 𝐿 then, its slope 𝑚
sub one is equal to six minus negative seven over negative nine minus negative
seven, which simplifies to negative 13 over two.
Now, let’s find the slope of
the line 𝑀. Its slope 𝑚 sub two will be
calculated as three minus one over 14 minus one, and this is equal to two over
13. Now, we can check if 𝑚 sub two
is equal to negative one over 𝑚 sub one. If we didn’t know the value of
𝑚 sub two, we could find a perpendicular line to the line 𝐿 by taking 𝑚 sub
two and setting it equal to negative one over negative 13 over two. And this would indeed give us a
value of two thirteenths for 𝑚 sub two. We can therefore give the
answer that the statement which is true about the lines 𝐿 and 𝑀 is option
(B). They are perpendicular.