Video Transcript
In this lesson, what we’re going to
be doing is learning how to find the horizontal or vertical distance between two
points on the coordinate plane. And we can think of the vertical
distance as the change in 𝑦 and the horizontal distance as the change in 𝑥. So, by the end of this lesson, what
we hope you’ll be able to do is find the horizontal or vertical distance between two
points on the coordinate plane and find side lengths of shapes in the coordinate
plane.
And to enable us to do this, we
should already know how to identify coordinates in our coordinate plane. For example, if we take a look at
this point here on our coordinate plane, the coordinates would be three, three. And that’s because it’s three on
the 𝑥-axis and three on the 𝑦-axis. Okay, great! Now let’s take a look at our first
example.
Find the length of the line segment
𝐴𝐵.
Well, there’re a couple of ways
that we can approach this question. So first of all, the first method
is to just count the squares between the two points. And that’s because if we join 𝐴 to
𝐵, we’ve got line segment 𝐴𝐵. And if we count the squares between
them, it’s gonna be six units. And that’s because each square is
one length unit. And what this is is the vertical
distance between 𝐴 and 𝐵. So therefore, we could say that the
length of the line segment 𝐴𝐵 is six length units.
However, I did say that there was
another way that we could look at this. Well, the other way to look at this
is to look at the coordinates for each point. So the coordinator of 𝐴 are six,
two. And the coordinates of 𝐵 are six,
eight. And because they have the same
𝑥-coordinate, this means that we know that they’re in the same vertical plane. So therefore, we know that the
distance between them is just gonna be the change in 𝑦, so the change in their
𝑦-coordinates. And the change in their
𝑦-coordinates is just gonna be eight minus two, which is equal to six, so it gives
us the same answer.
So this was a nice, simple example
to start off. But what we’re gonna have a look at
now is another example that involves both the horizontal and vertical distances
between points. And so far, we’ve shown a couple of
methods to do this. But what we’re gonna do before we
have a look at the next example is have a look at the distance between two points
formula. So the distance between two points
formula is something we can use to find the distance between any two points on our
coordinate plane. So let’s consider two points. So the two points we’ve got are 𝐴,
which we can denote with 𝑥 sub 𝐴, 𝑦 sub 𝐴, and 𝐵, which is 𝑥 sub 𝐵, 𝑦 sub
𝐵.
Well, in fact, what we can do is
find the distance between them by applying the Pythagorean theorem. And that is that we could say the
distance between 𝐴 and 𝐵 is equal to the square root of 𝑥 sub 𝐵 minus 𝑥 sub 𝐴
all squared, so the change in 𝑥 all squared, plus 𝑦 sub 𝐵 minus 𝑦 sub 𝐴 all
squared, so that’s the change in 𝑦 all squared. And the reason this works is if we
imagine two points, said point 𝐴 and point 𝐵, well, then, if we join them
together, then what we’d have is a right triangle with the vertical length being the
change in 𝑦, so 𝑦 sub 𝐵 minus 𝑦 sub 𝐴, and the horizontal length being the
change in 𝑥, 𝑥 sub 𝐵 minus 𝑥 sub 𝐴.
So this method could be used to
find the distance between any two points on the coordinate plane. However, we’ve already seen that if
we’re looking at horizontal or vertical distances, we can find it using more simple
methods. But what we do want to do is
demonstrate how this would in fact work for problems that do involve horizontal and
vertical distances. So we’re gonna use it in our next
example.
Find the lengths of the line
segment 𝐴𝐵 and the line segment 𝐷𝐶, where the coordinates of points 𝐴, 𝐵, 𝐶,
and 𝐷 are negative two, three; five, three; negative two, negative four; and
negative two, negative five, respectively, considering that a length unit is equal
to one centimeter.
So the first thing we’ve done is
marked our line segments onto our diagram, so we’ve got 𝐴𝐵 and 𝐷𝐶. And as we actually have a
horizontal line for 𝐴𝐵 and a vertical line for 𝐷𝐶, then what we can do is use a
couple of methods to solve the problem. The first method is the most
straightforward method, but I also want to show you a more formalized method just to
show you how a formula could be used for this type of problem as well. Well, in order to find the length
of the line segment 𝐴𝐵, because it’s horizontal, what we need to do is look for
the change in the 𝑥-coordinates.
So therefore, we can find this by
having the 𝑥-coordinate of 𝐵 and subtracting from it the 𝑥-coordinate of 𝐴. So we have five minus negative two,
which will give us an answer of seven centimeters. And we can check this by counting
on the squares on our diagram. So we got here seven, so it would
be seven centimeters.
Okay, great! So now let’s have a look at the
line segment 𝐷𝐶. Well, this time as we’re looking at
the length of the line segment 𝐷𝐶, which is a vertical line, we’re gonna be
looking at the change in our 𝑦-coordinates. So what we’re gonna have is
negative four minus negative five, which will give us the answer one. So we know that it’s one centimeter
long. And again, we could check that out
by counting the squares that we have on our axis, knowing that every two squares is
equal to one unit or one centimeter.
It is worth addressing at this
point what would happen if you took the 𝑦-coordinates the other way around, so we
had negative five minus negative four. Well, this would give a result of
negative one. And what we can do is disregard the
negative, and that’s because we’re only interested in the magnitude because we’re
looking at distance. So therefore, we would just get
one, which would be one centimeter as well. Now, the reason we have it the way
around that we have in the question and in our answer is because to calculate how
far it is from 𝐷 to 𝐶, we usually start with the coordinates of 𝐶 and then
subtract the coordinates of 𝐷.
So, we’ve solved the problem as we
said; however, what we’re also gonna look at is a formalized way of doing it using
the distance formula. And what the distance between two
point formula tells us is that the distance between 𝐴 and 𝐵 is equal to the square
root of 𝑥 sub 𝐵 minus 𝑥 sub 𝐴 all squared plus 𝑦 sub 𝐵 minus 𝑦 sub 𝐴 all
squared. And this is from the Pythagorean
theorem.
Well, to demonstrate how this would
work for our 𝐴𝐵, so our line segment 𝐴𝐵, this would be equal to the square root
of five minus negative two all squared, so that’s the change in our 𝑥-coordinates,
plus three minus three all squared, which would just give us the square root of
seven squared. And that’s because for the second
part, we’ve got three minus three, which is just zero, and zero squared is just
zero, which would just be the square root of 49. So it’d just be seven. Also, we know if we have the square
root of a squared number, then it would just be the number itself. So great! That’s 𝐴𝐵.
So now let’s have a look at 𝐷𝐶
using this method. So then for 𝐷𝐶, what we’re gonna
have is square root of negative two minus negative two all squared plus negative
four minus negative five all squared, which would just give us root one squared. So this would just give us root
one, which just gives us the answer of one. So we’ve got the same answer as
before. It is worth noting that we’ve got
root 49 and root one. It would usually have two results,
positive or negative, so positive or negative seven or positive or negative one. However, in this instance, we’re
just looking at the positive result because we’re looking for a length or a
magnitude.
Okay, so we’ve now looked at an
example that was finding horizontal and vertical lengths, and we’ve also looked at
the distance formula. What we’re gonna take a look at now
is a problem that involves a shape.
Find the length of the base of
triangle B.
So the first thing we want to do is
check out the scale of our axes. And we can see that every square is
equal to one unit, and that’s because we have two squares is equal to two units,
four squares is equal to four units, etcetera. Therefore, if we count along the
base of our triangle, we can see that the length of it is going to be five length
units long. And it is also worth noting that we
could’ve seen the vertices of the bottom of our triangle and found out what their
𝑥-coordinates were, so in this case, it’s five and 10. Then we could’ve found the
difference between them. So 10 minus five gives us five,
which would have given us the same result.
So in this question, we just looked
at a simple shape. But what we’re gonna do is develop
this further now and look at perimeter and area based upon using these same
skills.
In a house design that uses a
coordinate plane, the vertices of a living room are plotted at two, two; two, eight;
nine, eight; and nine, two, where the coordinates are measured in meters. Determine the perimeter and the
area of the room.
So the first thing we’ve done here
is drawn a sketch of the scenario. So we have the four vertices
plotted. And then we can join these up to
make our living room. So now what we want to do is find
the lengths of our sides. So we can see that our living room
has horizontal and vertical sides. We know that because, for instance,
if we look at the top line, if we look at the vertices and their coordinates at
either end, we’ve got the same 𝑦-coordinate. So therefore, this is going to be a
horizontal line. If we look to the far right, then
we’ve got the same 𝑥-coordinate. So this is going to be a vertical
line.
So if we want to work out the
length of the top line, what we’re gonna do is find the difference between the
𝑥-coordinates. So we have nine minus two, which is
gonna be equal to seven. If we hadn’t drawn the diagram, we
could still see this because if we look at our points that we’ve been given, we
could see that the two that have the same 𝑦-coordinate here are two, eight; nine,
eight. So therefore, we find the
difference between their 𝑥-coordinates, again, which would give us seven. And we know that the units are in
meters.
Now, what we’d expect by looking at
the diagram is that the distance between our point two, two and nine, two would be
the same because it does look like we have a rectangle. Let’s double-check this. Well, yes, once again, if we look
at the 𝑥-coordinates and the difference between them, we have nine minus two, so
it’s gonna give us seven. So we know that’s seven meters long
as well.
So now let’s take a look at our
vertical sides. Well, for our vertical sides, what
we’re looking for are the pairs of coordinates who have the same 𝑥-coordinates. So we’ve got nine, eight; nine, two
and two, eight; two, two. And then what we want to do is find
the difference between their 𝑦-coordinates. And here it’s gonna be eight minus
two for both, which is gonna give us six. So therefore, we know the height is
six meters.
And as we’ve got two pairs of equal
parallel sides, we know that what we’re looking at here is a rectangle. So now what we can do is find the
perimeter and the area. Well, the perimeter is the distance
around the outside, so it can be seven add six add seven add six, which is gonna
give us an answer of 26 meters. But then if you want to find the
area, the area is equal to the length multiplied by the width. So it’s gonna be equal to seven
multiplied by six, which is gonna give us an area of 42 meters squared.
So now, for our final example, what
we’re gonna do is look at a problem where we have to form a shape and then find the
perimeter of that shape.
Given that the coordinates of
points 𝐶 and 𝐷 are seven, five and seven, four, respectively, find the perimeter
of the figure 𝐴𝐵𝐶𝐷.
So the first thing we’ve done is
marked our points 𝐶 and 𝐷 onto our diagram. And then what we can do is join our
points to create our figure 𝐴𝐵𝐶𝐷. Now, what we can do is check out
the axes. And in our axes, we can see that
every square is equal to one unit. So therefore, we can count from 𝐵
to 𝐶. This is three units. 𝐶 to 𝐷 is one unit. 𝐷 to 𝐴 is three units. And 𝐴 to 𝐵 is one unit. So therefore, as we’ve got two
pairs of equal parallel sides, we can see that this is in fact a rectangle.
So to find the perimeter, what we
want to do is find the distance around the outside. And the simplest way of doing that
is by adding three, one, three, and one. And this gives us a perimeter of
eight. So what we’ve done is we formed the
figure 𝐴𝐵𝐶𝐷 and found its perimeter.
So now we can see that we’ve looked
at various different examples, some involving shapes, some involving straight line
segments. We’ve looked at how we’d find the
difference between different coordinates. We looked at how we can just find
perimeter, for instance, by counting squares. And we’ve also looked at the
distance formula. So now let’s have a quick summary
of the lesson.
Well, the main key points from this
lesson are if we want to find the length of a horizontal line on the coordinate
plane, then we can find this by finding the difference between two 𝑥-coordinates at
either end, so 𝑥 sub two minus 𝑥 sub one. If we want to find the vertical
length of a line on the coordinate plane, then this is the difference between the
𝑦-coordinates at either end of the line, so 𝑦 sub two minus 𝑦 sub one.
What we’ve also looked at is the
distance formula and how if we want to find the distance between two points 𝐴 and
𝐵, then this is equal to the square root of 𝑥 sub 𝐵 minus 𝑥 sub 𝐴 all squared,
which is in fact the change in our 𝑥-coordinates all squared, plus 𝑦 sub 𝐵 minus
𝑦 sub 𝐴 all squared, which is the change in our 𝑦-coordinates. And we get this from the
Pythagorean theorem. So what we’ve done is explored
these two methods and also looked at counting squares or counting units using our
coordinate axes. And all three methods are methods
we can solve problems of this type.
