The coordinates of points 𝐴, 𝐵,
𝐶, and 𝐷 are three, two; negative one, seven; three, one; and nine, two,
respectively. Are the line segments 𝐴𝐵 and 𝐶𝐷
So to help us actually understand
what’s going on in this question, I’m just gonna do a little sketch. So I’ve sketched some axes
here. And now, I’m gonna sketch on our
four points. And we actually need this to answer
this question just so we can actually get a visual picture of the question.
So first of all, we got point 𝐴 at
three, two, so a three on the 𝑥-axis, two on a 𝑦-axis. And then, we have point 𝐵 at
negative one, seven, point 𝐶 at three, one, and finally point 𝐷 at nine, two. Okay, so now, we got our four
points. But what the question is actually
looking for is are the line segments 𝐴𝐵 and 𝐶𝐷 perpendicular.
So we now got our line
segments. Let’s see if they’re
perpendicular. So first of all, what does
perpendicular mean? Well, if two lines were actually
perpendicular to one another, it means they’re actually at right angles. So that’s what the word
But how would we know if two lines
are actually perpendicular to one another? Well, actually, we have a special
relationship between perpendicular lines and it has to do with the slopes, where the
slope is actually- well we call it 𝑚. So if we got the slopes of two
perpendicular lines and we call them 𝑚 one and 𝑚 two, then if we multiplied them
together we get a result of negative one.
So another way of thinking about it
is actually the slope of one line is equal to the negative reciprocal of the slope
of a line perpendicular to it. So it’s negative one over the slope
of that other line. Okay, so bearing that in mind,
let’s now find the slope of 𝐴𝐵 or 𝐵𝐴 and 𝐶𝐷.
And to actually help us find the
slope of our two subsections or two lines, we have the slope formula. And it tells us that the slope 𝑚
is equal to 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. And what this actually means is the
change in 𝑦 divided by the change in 𝑥, so it’s the change in our 𝑦-coordinates
divided by the change in our 𝑥-coordinates.
Okay, so now, we got everything we
need to solve the problem. So for line 𝐴𝐵, we’ve got 𝐴,
which is three, two, so coordinates three, two. I’ve actually labelled these 𝑥
one, 𝑦 one and then 𝐵 which is negative one, seven. And I’ve labelled this 𝑥 two, 𝑦
So therefore, using our formula, I
can say that the slope or I’ve called it 𝑚 one in this case cause it’s the first
one we’re calculating is equal to seven minus two because that’s our 𝑦 two minus
our 𝑦 one divided by our negative one which is our 𝑥 two minus three because three
is our 𝑥 one. So this is gonna give us an answer
of negative five over four and that’s because seven minus two is five; negative one
minus three is negative four. So therefore, we’ve made the whole
fraction negative, so it’s negative five over four or negative five-fourths.
Okay, so that’s 𝑚 one. So that’s the slope of 𝐴𝐵. Now, let’s find the slope of
𝐶𝐷. So now, for 𝐶𝐷, we’ve got 𝐶 is
three, one, so 𝑥 one, 𝑦 one. And 𝐷 is nine, two. So that’s 𝑥 two, 𝑦 two. So therefore, what we’re gonna get
is two minus one, that’s cause that’s 𝑦 two minus 𝑦 one, divided by nine minus
three because that’s our 𝑥 two and our 𝑥 one. So therefore, we’ve got a result as
one over six. We’re gonna say that the slope of
𝐶𝐷 is one over six or one-sixth.
So now, what we’re gonna do is
actually use the first relationship we talked about, which was that 𝑚 one
multiplied by 𝑚 two. So if two lines are actually
perpendicular to each other, then the result is gonna be negative one. So we’re gonna use this to see
whether our two lines are actually perpendicular, so 𝑚 one multiplied by 𝑚
Well, this is gonna give us
negative five over four, so negative five-fourths or five-quarters, multiplied by
one-sixth, which is equal to negative five over 24. And that’s cause you multiply the
numerators and multiply the denominators.
Well, negative five over 24 is not
equal to negative one. So therefore, we can say that line
segments 𝐴𝐵 and 𝐶𝐷 are definitely not perpendicular to one another.