WEBVTT

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The gradient of a line is
a measure of how steep

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that line is.

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We may have a
very steep line like

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that.

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And the gradient will
be larger than a line.

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Which is a bit more shallow.

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So lines that are fairly shallow
like this one will have fairly

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Lopes, fairly small gradients,

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steep lines. Have large
gradients and lines.

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Which are horizontal, will
have zero gradients?

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And we need to try to quantify
that a little bit. Try and do

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this a little bit more
mathematically so we can

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actually measure how much
steeper this is than this one

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than this one, and so on.

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I'd run three line
segments on this

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diagram. Let's
just look at them.

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The first one.

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Is the line segment from A to D?

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Now as we move from A to D.

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The X coordinate increases from
one to two.

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And the Y coordinate increases
from one to five.

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Now the line segment AD is
steeper than the line segment

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AC. As we move from A to C.

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Exchange is from one to two and
Y changes from one to three.

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An AC in turn is steeper than
the line segment AB. The line

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segment AB is in fact
horizontal because as X

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increases from one to two, the
Y coordinate doesn't change at

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all. It remains at one.

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Let's try and think about why
mathematically, the line AD is

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steeper than the line AC.

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And the reason for this is that
in both cases are X coordinate

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is changing from one to two.

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But as we move from A to D,
there's a much bigger.

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Change in Y than if we move from
A to see so it's this relative

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change in Y relative change in
X, it's going to be important.

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What we do is we calculate the
change in Y.

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Divide it by the
change in X.

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That's going to be a measure of
the steepness. Let's do it for

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the point AD for the points A&E.

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OK, as we move from A to D.

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Why changes from one to five? So
the change in Y?

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Is 5 - 1 and the change in X
while exchanges from one to two.

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So the change in X is 2 - 1.

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So this quantity, the change
in Y over change in X for the

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line AD is 5 - 1, which is
four 2 - 1 which is one and

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four over one is 4.

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So that's a measure of how much
why changes as exchanges.

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From one to two. What about
the segment AC? Let's do

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the same thing.

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While the change in Y now is
from one to three.

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So the changes 3 - 1.

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The change in X well X goes from
one to two, so the changes 2 - 1

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and again 3 - 1 is two. 2 - 1
is one and 2 / 1 is 2. So this

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is a measure of the relative
change in X&Y.

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What about a bee?

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Well, as we move from A to B,
why doesn't change at all? So

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the change in Y is 1 - 1, which
is of course 0.

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And the change in X is still
2 - 1, so we get zero over

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one which is 0.

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So you see this quantity change
in Y divided by changing X gives

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us a measure of the steepness of

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these lines. As we would expect,
the change in Y over change in X

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for AD. Which we turned out to
be 4 is greater than the change

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in Y over change in X for AC
because ady is steeper than AC

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an intern. This change in Y over
change in X for AC.

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Is greater than the change in Y
over change in X for a bee

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because AC is steeper than a B?

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So it's this quantity which
gives us the measure that we're

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looking for, and it's this
quantity we define to be the

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gradient of the line segment.

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We often use the symbol M for

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gradient. So the gradient is
defined to be the change

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in Y. Divided by
the change in X.

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As we move from one point to a

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neighboring point. Let's do that
for some general case.

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Suppose we have.

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System of coordinates

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appoint a. X1Y
One.

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And a point B.

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X2Y2 And
we're interested in the gradient

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of the line segment joining A&B.

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Let me put in a horizontal line

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through way. Anna vertical line.

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Through be. So there's my X
axis. There's my Y Axis.

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As we move from A to B.

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Exchange is from X one.

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2X2.

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Why changes from Y1?

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To Y2.

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So the change in Y divided
by the change in X.

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While the change in Y is the
final value minus the initial

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value, so it's Y 2 minus Y 1.

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The change in X is X2
minus X one.

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And that is the formula that we
can always use to find the

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gradient of the line joining two

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points. We can think of this

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another way. Suppose we look at
this angle in here. Let's call

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that angle theater.

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Now the change in Y is
just this distance here.

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Distance in there.

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And the change in X is this
distance in here.

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And if we take the change in Y
and divide it by the change in

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X, what we actually get is the
ratio of this side of this right

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angle triangle to this side. And
that's just the tangent of this

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angle here. So this quantity
that we've calculated is not

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only the gradient of the
line, it's also the tangent

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of the angle that the line
makes with the horizontal.

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So the gradient M which we said
is Y 2 minus Y, one over X2

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minus X one is also equal to the
Tangent Theta, where Theta is

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the angle that the line makes
with the horizontal.

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We can take this a stage
further. Suppose we continue

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this line backwards until we

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meet. The X axis.

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And this angle in here between
the extended line and the X

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axis. Corresponds to this
angle. Here these are

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corresponding angles, so this
two must also be theater.

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So In other words, the gradient
of the line is also the tangent

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of the angle that the line makes
with the X axis.

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Let's have a
couple of examples.

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Let's choose a couple of points.
Supposing a is the .34.

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And B is the point.

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814. Let's
calculate the gradient of the

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line joining these two points.

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Well, the gradient is simply the
difference in the Y coordinates

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14 - 4.

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Over the difference in the X

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coordinates. 8 - 3.

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14 - 4 is 10 and 8. Subtract
3 is 5 and 5 into 10 goes

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twice. So the gradient of this
line is 2.

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A second other example, suppose
we have the point a, which

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now has coordinates 04 and B
which has coordinates 50. Let's

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do the same calculation.

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The gradient will be the
difference in the Y coordinates.

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That's 0 - 4.

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Divided by the difference in the
X coordinates 5 - 0.

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So this time will get minus four
on the top, five. At the bottom

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we get minus four fifths.

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So this is a little bit
different now because we found

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that we've got a negative number

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for our gradient. And see
what that actually means.

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Let's plot the points and
see what's going on.

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Point A0X coordinate Y
coordinate of four. So let's put

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that there that's Point A.

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Point B has an X coordinate of
five that's there.

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Y coordinate of 0, so there's my
point there in there.

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And the line joining them looks

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like this. We know that
this line has gradient

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minus four fifths.

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This line, as you notice, is
sloping downwards as we move

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from left to right.

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And that's why the gradient
turns out to be negative.

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Another way of thinking about
this is that the angle that the

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line now makes with the X axis.
This angle in here this theater

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is now an obtuse angle.

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Greater than 90 degrees less
than 180 degrees, so we've an

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obtuse angle. A line which is
sloping downwards from left to

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right. And a negative gradient.

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Let me try to summarize
all that behavior. If

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you have a situation.

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Like this?

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Where the angle that the
line makes with the

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horizontal is acute.

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Then the gradient.

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Will be positive.

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And the reason for that is that
as you move along the line.

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As X increases, why also
increases so the change

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in Y and the change in X
have the same sign.

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It's also important to recognize
that if we take the tangent of

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an acute angle, you get a
positive number, so Tan Theater,

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which we know is the same as
them, is also positive.

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What about an angle that
sloping alignment sloping

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downwards? We know that the
angle is now.

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Theater and it's obtuse.

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We know that the tangent of an
obtuse angle is negative, and as

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we've seen, the gradient is

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negative. And that's be'cause
as X increases.

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Why is decreasing so the change
in Y and the change in X have

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different signs, so we take the
ratio, will find out that the

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gradient is actually negative.

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And finally.

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Let's have one where theater is
0, so the angle that the line

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makes with the horizontal is 0,
while tan feta is 0. So that's

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consistent with our intuition.
That tells us that MSO the

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gradients 0. Let's
have a look

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at some parallel

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lines.
Here's

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a

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line.
Let's call it L1.

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L1 will make a certain angle.

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Theater one with the X axis.

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So it's gradient, as we've seen
already, is Tampa Theatre 1.

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M1, its gradient is tan.

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Theater.

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Let's put another line on this,
also parallel to this first

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line. This line is L2.

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It will have a gradient M2.

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That's extend it back to the.

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Horizontal axis And let's
measure this angle that would be

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theater 2. And M2 will be the
tangent of Theta 2.

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Now, because these two
lines are parallel.

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They cross this X axis at the
same angle, Theta one and three

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to two. A corresponding angles
Sophie to one must be equal to

00:14:34.960 --> 00:14:35.974
three to two.

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So because the to one is 3 to 2.

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10 three to one what he called
10 theater 2 so. In other words,

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M1 equals M2.

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So for two parallel lines,
as you might have expected,

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intuitively, the two
gradients are equal.

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And Conversely, if we have two
lines for which the gradients

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are equal. Then we can deduce
from that that the two lines

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must be parallel. OK, so that's
parallel lines. Let's look at

00:15:12.328 --> 00:15:14.251
some perpendicular lines.

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See if we can do something
about the gradients of

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perpendicular lines.

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Start with a point P.

00:15:23.790 --> 00:15:29.675
And the origin there. And let's
suppose point P has coordinates

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a speed. That means that to get
to pee from oh, we go a

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distance a in the X direction
and be in the Y direction.

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Now what I'm going to do now is
I'm going to draw a

00:15:48.098 --> 00:15:50.858
perpendicular line, a line that
is perpendicular to Opie, and

00:15:50.858 --> 00:15:54.170
I'm going to do that by taking
opian, rotating it through 90

00:15:54.170 --> 00:15:57.482
degrees. So the point P will
move around here and it will

00:15:57.482 --> 00:15:58.862
move to appoint up there

00:15:58.862 --> 00:16:00.924
somewhere. And let's call that

00:16:00.924 --> 00:16:04.505
new point Q. I'm going to
try to figure out what the

00:16:04.505 --> 00:16:05.240
coordinates of QR.

00:16:06.780 --> 00:16:12.032
This angle hit in here is 90
degrees because Opie and oq are

00:16:12.032 --> 00:16:17.141
perpendicular. Now to get from O
to pee, wee had to go

00:16:17.141 --> 00:16:20.851
horizontally a distance A and
vertically be. So if this

00:16:20.851 --> 00:16:25.303
triangle shifts around over here
to get from Otak you will have

00:16:25.303 --> 00:16:27.529
to go vertically a distance a.

00:16:30.800 --> 00:16:32.505
And then horizontally a distance

00:16:32.505 --> 00:16:36.133
be. So you see, we've just
shifted this triangle, rotated

00:16:36.133 --> 00:16:40.124
it round through 90 degrees, and
doing that we can then read off

00:16:40.124 --> 00:16:41.659
the coordinates of Point Q.

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Q will have an X coordinate of

00:16:44.670 --> 00:16:50.880
minus B. And AY
coordinate of A.

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That's now calculate the
gradient of the line opi. Let's

00:16:56.700 --> 00:17:02.233
call that MOPMLP. Remember, is
the change in Y divided by

00:17:02.233 --> 00:17:05.754
changing X as we move from OTP.

00:17:06.370 --> 00:17:12.670
As we move from outer P, the
change in Y is B minus zero.

00:17:12.670 --> 00:17:18.166
The change in X is
a minus zero.

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So the gradient of Opie is
just be over A.

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What about the gradient of
OQ? Let's call that MOQ.

00:17:33.040 --> 00:17:36.910
We want the change in Y divided
by the change in X as we move

00:17:36.910 --> 00:17:37.942
from O to Q.

00:17:39.060 --> 00:17:46.230
Well, the change in Y as we move
from outer Q is a subtract 0.

00:17:46.350 --> 00:17:51.900
And the change in X is minus
B, subtract 0.

00:17:51.900 --> 00:17:55.977
So this time this
simplifies to a over minus

00:17:55.977 --> 00:17:58.695
B or minus a over B.

00:18:00.780 --> 00:18:05.301
Now let's see what happens when
we multiply these two results

00:18:05.301 --> 00:18:09.822
together. Let's take MOP and
we're going to multiply it by

00:18:09.822 --> 00:18:17.200
MCU. That's be over a
multiplied by minus a over B.

00:18:17.740 --> 00:18:22.492
And you see, when we do that,
the aids cancel the beast

00:18:22.492 --> 00:18:24.472
cancel, and we're left with

00:18:24.472 --> 00:18:25.700
just. Minus one.

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This is a very important result
if you have two perpendicular

00:18:31.275 --> 00:18:35.730
lines, then the product of their
gradients is always minus one.

00:18:36.900 --> 00:18:40.123
And correspondingly, if you've
got 2 lines and you find that

00:18:40.123 --> 00:18:43.346
when you multiply the gradients
together, you get minus one, you

00:18:43.346 --> 00:18:46.276
can deduce from that that the
lines must be perpendicular.

00:18:47.720 --> 00:18:50.912
Let's just have a look at an

00:18:50.912 --> 00:18:53.705
example. Let's have three

00:18:53.705 --> 00:18:57.098
points. Using A is the

00:18:57.098 --> 00:19:00.532
.12. Because the

00:19:00.532 --> 00:19:04.526
.34. And see is the point is

00:19:04.526 --> 00:19:10.520
not 3. And we'll ask ourselves,
the question is AB.

00:19:10.520 --> 00:19:11.460
Perpendicular

00:19:12.530 --> 00:19:17.824
To AC. Question is a
be perpendicular to AC?

00:19:18.480 --> 00:19:23.364
Well will will do this by
calculating the gradient of the

00:19:23.364 --> 00:19:26.472
line from A to B. Let's call

00:19:26.472 --> 00:19:31.068
that MAB. And then we'll find
the gradients of the line from A

00:19:31.068 --> 00:19:32.916
to C will call that Mac.

00:19:33.490 --> 00:19:37.594
So let's do this calculation. We
want the gradient of the line

00:19:37.594 --> 00:19:41.698
from A to B. Well, that's simply
the difference in the Y

00:19:41.698 --> 00:19:43.066
coordinates 4 - 2.

00:19:43.110 --> 00:19:47.200
Over the difference in the X
coordinates 3 - 1.

00:19:47.200 --> 00:19:53.518
4 - 2 is two 3 - 1 is 2, so
the gradient of a B is one.

00:19:54.880 --> 00:19:56.518
What about the gradient from A

00:19:56.518 --> 00:20:02.974
to C? Well, the difference in
the Y coordinates now is 3 - 2.

00:20:03.000 --> 00:20:07.480
The difference in the X
coordinates is 0 - 1.

00:20:07.810 --> 00:20:14.530
So we've got 3 - 2 is one
0 - 1 is minus one.

00:20:14.540 --> 00:20:16.916
So all this simplifies
which is minus one.

00:20:18.230 --> 00:20:23.280
If we multiply the two gradients
together, maybe multiplied by

00:20:23.280 --> 00:20:26.310
Mac, will get one times minus

00:20:26.310 --> 00:20:31.328
one. Which is clearly minus one,
so the gradients of these two

00:20:31.328 --> 00:20:35.816
lines multiplied together have a
result which is minus one, and

00:20:35.816 --> 00:20:39.896
that means that the two lines a
be an AC.

00:20:40.410 --> 00:20:42.118
Indeed, must be perpendicular.