www.mathcentre.ac.uk/.../Properties%20of%20straight%20line%20segments.mp4

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.