In this unit, we're going to have a look at two points.
And the line joining the two
points. We call this a straight line segment and I'm going to
show you how we can calculate the distance between the two
points, or In other words, the length of this line segments and
how we can calculate the coordinates of the midpoint of
the line segment.
Now, before we do that, I'd like to revise some facts about the
coordinates of a point.
Suppose we choose a point, let's call this .0.
And suppose we draw a horizontal
line. Through that point.
And a perpendicular line align at 90 degrees also
through that point.
We call these two perpendicular lines axes.
And this horizontal line. We'll call it the X axis and label it
with the letter X.
And the vertical line.
Will call this the Y axis and label it with the letter Y.
Now on these axes we draw a scale and what we do is we
draw what's called a uniform scale. That means it's evenly
spread along the axis.
And so on and on the Y axis we do. Likewise, we put a uniform
scale on the Y axis.
And the point of doing that is that we can then measure
distances in the X direction and in the Y direction.
Now suppose we choose any point at all in the plane.
Let's choose a point. Here, let's call this point A.
What we can do is we can measure the distance of a.
From the vertical axis, so we
can measure. That distance in
there. And because that's the distance in the X direction,
we call it the X coordinate of the point a.
We can also measure the distance from the horizontal axis.
And because that's measured in the Y direction, we call that
the Y coordinate of A.
And these two numbers, the X coordinate and the Y coordinate
we write in brackets after the letter A itself.
And we call XY the coordinates of the point a.
We also refer to XY in the brackets like. This is an
ordered pair of numbers, is clearly a pair of numbers, and
it's an ordered pair because the order is very important, we put
the X coordinate first. That's the distance from the Y axis.
And the Y coordinate second, that's the distance from the X
axis. So now we have a way of specifying precisely
where in the plane our point of interest lies.
If we move to the left hand side of the Y axis, then what we have
are negative X coordinates because as we move to the left.
We count down from zero, so minus 1 - 2 - 3 and so on. So
anything to the left of the Y axis will have a negative X
coordinate. And similarly, if we want to come down below the
horizontal axis, we want to come down here. We countdown
from 0 - 1 - 2 - 3 and so on. So any point with a
negative Y coordinate will be below the horizontal axis.
Let's plot some points and see where they lie.
Not prepared, some graph paper for us.
So here's my piece of graph paper. You can see my X&Y Axis
and my uniform scales drawn on the two axes, and let's plot
some points. Here's the first Point, Point A and let's suppose
point a has coordinates 3. Two. Let's see where that is.
Well, a has an X coordinate of three, so we move in the X
Direction a distance 3.
And AY coordinate of two, so we're moving up from the X axis
of distance to and that takes us to this point here.
X coordinate of three Y coordinate of two.
Let's have another point. Let's have point B.
Switch those Round 2 three. Let's see where that is now
with an X coordinate of two. So we've got to move a
distance to from the Y axis.
And AY coordinate of three means we move vertically. So
two across three up takes us to this point there. And
that's my point, B.
Let's have some more. What about this Point C, which has
an X coordinate of zero and a Y coordinate of three? Let's
see where that is.
When X coordinate of 0 means we move no distance at all in the X
direction, so we stay on the Y
axis. And we want to be a distance three upwards from the
origin. So there's my Point C.
Point D that suppose that's minus 3 zero. So I've introduced
a negative X cord in it. Now remember with a negative X
coordinate, we're moving to the left hand side of the Y axis. We
want to move a distance 3123, so we're on this line here.
And AY coordinate is 0 means we don't move up or down at all. So
at that point there. So there's
my point, D. Have a couple more E minus 2 - 3 so we've got a
both a negative X and a negative
Y value. Minus 2 on the X wings means we moved to the
left distance 2 - 3 on the Y means we move down a distance
three, so we're down there to get to point E.
Final one 2 - 2. Let's see where that is.
X coordinate X coordinate of two means. We moved to the writer.
Distance to. Y coordinate of minus 2
means we moved down a distance too, so
there's point F.
OK, so now given any ordered pair of numbers representing a
point in the plane, you know how to plot where it is. We
know how to locate it.
So now we know how to plot the points. Let's see if we can plot
two points and find the distance between the two points.
So some more graph paper here. And let's suppose we pick any
two points and let me suppose the first point, let's call it
a, has coordinates 13.
Let's put that on the graph.
X coordinate if one Y coordinates of three means we
are one across and three up, so at that point there that's a.
And let's choose my second point to be four 5X
coordinate of four Y coordinate of five puts me
there.
So two points.
Let's join them.
And that gives me my straight line segment and what I'm going
to do is I'm going to show you
how we can calculate. The length of this line segment, or In
other words, the distance between points A&B.
Now, the way we do this is as follows. We look at the
point A and will draw in a horizontal line through a.
So because this line is horizontal, it's parallel
to this X axis.
I'm going to draw in a vertical line through be.
And again, because that's a vertical line, it's parallel to
the Y axis, and so this angle in here must be 90 degrees, and we
see what we've done is. We've formed a right angle triangle.
Anywhere along this horizontal line has a Y
coordinate of three, so in particular the Y
coordinates at this point here, where the right
angle is must be 3.
Call this Point C. It's Y coordinate must be 3.
Now anywhere on this line.
Has an X coordinate of four the same as the Point B, so the X
coordinate here must be 4. So now we know the coordinates of
this point as well.
Let's put the coordinates of being 45 and a in 1 three.
Now what I'm going to do is I'm going to show you how we can
calculate this distance in here.
And this distance in here, because then we will know the
lengths of two of the sides of this right angle triangle.
And then we're going to use Pythagoras theorem to get
this length, because this length is the hypotenuse of
this right angle triangle.
Let's look at this distance here.
Now the total distance from the X axis up to point B.
Is 5. The Y coordinate of be.
So the distance all the way up there is 5.
The distance from the X axis up to Point C.
Is 3 the white coat?
You need to see so that distance is 3.
So this distance that we're really interested in here
must be the larger distance. Subtract the smaller distance
5, subtract 3.
Which is 2.
It's important that you recognize that we've calculated
this distance by finding the difference of the Y coordinates,
5, subtract 3.
What about this distance in here?
Well, the total distance from the Y axis to this
dotted line is full.
Which is the X coordinate of be or the X coordinate of C?
This shorter distance from the Y axis to Point A is just
one. That's The X coordinate of a, so that's one. So the
distance we're really interested in is the larger
minus the smaller 4 - 1.
And 4 - 1 is 3.
And again 4 - 1 is the difference of the two
X coordinates, 4 - 1, giving you the three.
So now we know the length of the base of this triangle is 3. We
know its height is too.
And we can use Pythagoras theorem to find a B.
Pythagoras theorem will say that AB squared.
Is the sum of the squares of the other two sides. AB squared will
be 3 squared +2 squared.
3 squared is 9. Two squared is four and nine and four is 30.
So AB squared is 13, so AB, which is the distance we
want is going to be just the square root of 13.
OK, so knowing the coordinates of the two points.
We can find the lengths of the sides of this triangle and then
use Pythagoras theorem to get the hypotenuse, which is the
distance between the two points.
Now that's all well and good for the specific case we looked at,
but what I want to do now is do it more generally, instead of
having specific points, I want to take two arbitrary points in
the plane and see how we can find the distance between the
two of them. So again, let me have a Y axis and X axis.
And and origin.
And let's pick our first point, let's call it A and let's
suppose it's got an arbitrary X, an arbitrary Y coordinate. So
let's call those X, one and Y one. So there's my point A.
I'm going to take a second point. Let's call that B.
And let's suppose its X coordinate is X2, Y
coordinate is. Why two? So these are arbitrary points.
Now could be anywhere.
We're going to try to find the distance between the two
points, or In other words, the length of this line segment.
As before, let's put in a horizontal line.
Through a.
And a vertical line through be.
To form this right angle
triangle. Here, let's call this Point C.
What are the coordinates of points, see?
Well, anywhere along this
horizontal line. Will have a Y coordinate, which is why one.
Because that's the same as the Y coordinate of a. So the Y
coordinate of C must be why one?
What's its X coordinate? Well, its X coordinate must be the
same as the X coordinate of bee which is X2. So X2Y one are the
coordinates of Point C.
Let's calculate this distance in here.
Now this distance.
Is the difference in the Y coordinates the Y coordinates
of B subtract the Y coordinate of C, so this is just Y 2
minus Y one. The difference in the Y coordinates?
Similarly here this distance.
Is the X coordinate of C which is X2 subtracted the X
coordinate of a which is X one. So this distance is simply X2
minus X one.
And now you'll see, we know the base and we know the height of
this right angle triangle and we can use Pythagoras theorem
to get the hypotenuse, which is the distance we require.
So Pythagoras theorem will say that AB squared.
The square of the hypotenuse is the sum of the squares of
the other two sides.
That's X2 minus X one all squared.
Plus Y 2 minus Y1 all squared.
And then if we want a B or we have to do is square root both
sides, so maybe will be the square root of X2 minus X one
all squared plus Y 2 minus Y one
or squared? And that is the general formula that we can
always use to calculate the distance between two arbitrary
points in the plane.
Let's look at a specific example. Suppose we want to find
the distance between the points AA, which has a coordinates
minus one and three and B which
has coordinates. Two and minus four, for example, how can we
use this formula to find the distance between these two
points without actually drawing all the points?
Well, using the formula AB will be the square root.
We want the distance, sorry. The difference between the X
coordinates. So that's 2 minus minus one.
All squared.
Plus the difference between the Y
coordinates, which is minus 4 - 3.
All squared.
So that's the square root of 2. Subtract minus one is 3 and 3
squared is 9.
Minus 4 - 3 is minus Seven and minus 7 squared is plus 49.
So finally AB will be the square root of 9 + 49.
Which is 58.
That's the distance between those two points.
We're not going to look at how you can calculate the
coordinates of the midpoint of a straight line segment. Before we
do that, let me just.
Consider this problem. Suppose we have an X axis.
And we have some points on the X axis. So suppose we have one
2345, and so on.
Let's choose two points on this X axis. Let's suppose we pick
this point. Where X is 2. Let's call that a.
And this point, where X is 4. Let's call that be and ask
yourself what is the X value.
At the midpoint of the line segment a be, so we're looking
at this segment Abe, and we want to know what The X value is at
the midpoint. I think it's fairly obvious from looking
at it that the X value at the midpoint is 3.
Another way of thinking of that is that 3.
Is the average of two and four.
Because if we average two and four Adam together
divide by 2, two and four is 6 over 2 which is 3.
So In other words, The X coordinates at the midpoint
of a line segment is the average of the X coordinates
at the two ends.
Just to convince you of that, let's have a look at
another problem.
So we have one 23456 and Seven. And let's suppose
this time we pick, say, the .3 for A.
And the point 74B.
And we're looking at this straight line. Segments here
between A&B. Again, it's rather obvious from looking at it.
The X value at the midpoint is 5.
And if we average the X values at the beginning and the end.
Of the line segment three added to 7.
Divided by two as the average of the X values is 10 over 2, which
is 5. So in a more general case, in a few minutes time will see
that when we want to find the coordinates of the midpoint of a
line, we just need to average the values at either end.
OK, let's look at a more general one. Suppose we have two points.
Suppose A is the .23, B is the
point. 45
Let's plot those. And then we're going to try to find the
coordinates of the midpoint of the line joining them.
OK. So as the .23, so here's Point A.
And Point B is 45, so be will be
up here. And about that.
And there's my line segment.
Maybe?
We are interested in the point which is the midpoint
of this line, so let's put that on there.
And let's call it point P.
And because it's the midpoint of the line, we
know that this length must be the same as this length.
AP Is the same length as PB.
Let's put some other lines in here.
A horizontal line through a.
Vertical line through be.
Horizontal line through P and a vertical line through P.
And let's label some of these points that we've created C.
And D. And he.
Now let's look at the line segment from A to E.
OK.
Sorry Peter.
Not step.
OK.
Yeah.
Now let's look at the triangle.
ACP. And the
triangle BD. Sorry, the triangle.
Yeah.
Now let's look at the triangle ACP.
And the triangle PD.
Be.
Both of these triangles are right angle triangles
because they both have a right angle here.
Both of the triangles have the same hypotenuse, which
has the same length 'cause they both have the same
length there in there.
Also, this angle in here.
Is the same as this angle in here because their corresponding
angles? So in fact, because we've got two angles and a
side in the first triangle corresponding to two angles
and the side in the second triangle, these two triangles
or what we call congruent.
Congruent triangles are identical in every respect, and
in particular. What this means is that this length from B down
to D. Most corresponding with the same as this length
from P down to see.
So In other words, because these two lengths are the same BD and
PC, which is the same as DE.
Then this point D.
Must be the midpoint of the line from A to B.
So the Y coordinate of D must be the average of
the Y coordinates at B&B.
Now the average of the Y coordinates at E&B is 3 + 5
/ 2, which is 8 over 2 which is 4. So the Y coordinates
at D must be 4.
What about the X coordinate?
If we look at this line from A
to C. Because the triangles are congruent, then this distance
from A to see must be the same as the distance from P to D,
which is the same as the distance from C to E.
So In other words, because these two distances are the same, see
must be the midpoint of AE.
And because C is the midpoint of
AE. It's X coordinate must be the average of the X coordinates
A&E. Now the average of those X coordinates is 2 + 4, which is 6
/ 2 which is 3. So the X coordinates here must be 3.
So the X coordinate is 3. We know the Y coordinate is 3. At
this point we decided that the Y coordinate was four and we know
the X coordinate is also 4.
So now we can read off what the coordinates of point PRP will
have an X Coordinate A3.
And AY coordinate of four.
Now that's very complicated. It seems very complicated, but in
practice it's very simple. Let's just look at these numbers
again. The X coordinate at the
midpoint. Which is 3 is just the average of the two X coordinates
2 + 46 / 2 is 3, the Y coordinate at the mid .4 is just
the average of the two Y coordinates. Five and three,
which is 8 / 2 is 4.
So in general, if we have a situation where we've got a
point AX1Y one and point BX2Y2.
The coordinates of the midpoint of the line AV are the average
of the X coordinates.
And the average of the Y coordinates.
Let's just look at a simple example just to
hammer that home.
Suppose we have a point A with coordinates 2 - 4 and be with
coordinates minus four and three. So two points and were
interested in the coordinates of the midpoint of the line AB.
Select this point BP.
The X coordinate will be the average of the two X
coordinates, which is 2. Added 2 - 4 over 2.
And the Y coordinate will be the average of the two Y
coordinates, which is minus four added to 3 / 2.
And if we tie do that up, will get 2 subtract 4, which is minus
2 over 2, which is minus one.
And minus 4 + 3 is minus 1 - 1 over 2 is minus 1/2.
So there is the coordinates of the midpoint of the line
joining amb.