0:00:01.960,0:00:05.344
In this unit, we're going to[br]have a look at two points.

0:00:07.730,0:00:09.590
And the line joining the two

0:00:09.590,0:00:14.530
points. We call this a straight[br]line segment and I'm going to

0:00:14.530,0:00:18.600
show you how we can calculate[br]the distance between the two

0:00:18.600,0:00:23.040
points, or In other words, the[br]length of this line segments and

0:00:23.040,0:00:26.740
how we can calculate the[br]coordinates of the midpoint of

0:00:26.740,0:00:27.850
the line segment.

0:00:28.540,0:00:32.258
Now, before we do that, I'd like[br]to revise some facts about the

0:00:32.258,0:00:33.402
coordinates of a point.

0:00:34.720,0:00:38.527
Suppose we choose a point, let's[br]call this .0.

0:00:39.790,0:00:42.232
And suppose we draw a horizontal

0:00:42.232,0:00:44.259
line. Through that point.

0:00:44.850,0:00:47.811
And a perpendicular line[br]align at 90 degrees also

0:00:47.811,0:00:48.798
through that point.

0:00:51.060,0:00:54.434
We call these two[br]perpendicular lines axes.

0:00:57.880,0:01:02.768
And this horizontal line. We'll[br]call it the X axis and label it

0:01:02.768,0:01:04.272
with the letter X.

0:01:05.060,0:01:06.380
And the vertical line.

0:01:07.700,0:01:11.756
Will call this the Y axis and[br]label it with the letter Y.

0:01:13.460,0:01:17.772
Now on these axes we draw a[br]scale and what we do is we

0:01:17.772,0:01:20.852
draw what's called a uniform[br]scale. That means it's evenly

0:01:20.852,0:01:22.084
spread along the axis.

0:01:25.580,0:01:30.170
And so on and on the Y axis we[br]do. Likewise, we put a uniform

0:01:30.170,0:01:31.700
scale on the Y axis.

0:01:34.480,0:01:38.188
And the point of doing that is[br]that we can then measure

0:01:38.188,0:01:41.278
distances in the X direction and[br]in the Y direction.

0:01:42.690,0:01:46.012
Now suppose we choose any[br]point at all in the plane.

0:01:46.012,0:01:49.032
Let's choose a point. Here,[br]let's call this point A.

0:01:50.470,0:01:54.358
What we can do is we can measure[br]the distance of a.

0:01:54.970,0:01:56.920
From the vertical axis, so we

0:01:56.920,0:01:59.176
can measure. That distance in

0:01:59.176,0:02:03.705
there. And because that's the[br]distance in the X direction,

0:02:03.705,0:02:07.455
we call it the X coordinate[br]of the point a.

0:02:09.360,0:02:12.560
We can also measure the distance[br]from the horizontal axis.

0:02:15.060,0:02:18.118
And because that's measured in[br]the Y direction, we call that

0:02:18.118,0:02:19.508
the Y coordinate of A.

0:02:20.930,0:02:24.725
And these two numbers, the X[br]coordinate and the Y coordinate

0:02:24.725,0:02:27.830
we write in brackets after the[br]letter A itself.

0:02:30.390,0:02:34.740
And we call XY the coordinates[br]of the point a.

0:02:36.370,0:02:40.066
We also refer to XY in the[br]brackets like. This is an

0:02:40.066,0:02:43.454
ordered pair of numbers, is[br]clearly a pair of numbers, and

0:02:43.454,0:02:47.150
it's an ordered pair because the[br]order is very important, we put

0:02:47.150,0:02:50.538
the X coordinate first. That's[br]the distance from the Y axis.

0:02:51.200,0:02:55.017
And the Y coordinate second,[br]that's the distance from the X

0:02:55.017,0:02:59.244
axis. So now we have a way[br]of specifying precisely

0:02:59.244,0:03:02.358
where in the plane our point[br]of interest lies.

0:03:03.790,0:03:08.798
If we move to the left hand side[br]of the Y axis, then what we have

0:03:08.798,0:03:12.241
are negative X coordinates[br]because as we move to the left.

0:03:14.500,0:03:19.796
We count down from zero, so[br]minus 1 - 2 - 3 and so on. So

0:03:19.796,0:03:24.099
anything to the left of the Y[br]axis will have a negative X

0:03:24.099,0:03:29.080
coordinate. And similarly, if[br]we want to come down below the

0:03:29.080,0:03:32.780
horizontal axis, we want to[br]come down here. We countdown

0:03:32.780,0:03:38.700
from 0 - 1 - 2 - 3 and so[br]on. So any point with a

0:03:38.700,0:03:42.030
negative Y coordinate will be[br]below the horizontal axis.

0:03:43.090,0:03:45.313
Let's plot some points[br]and see where they lie.

0:03:49.460,0:03:51.504
Not prepared, some[br]graph paper for us.

0:03:55.770,0:04:00.151
So here's my piece of graph[br]paper. You can see my X&Y Axis

0:04:00.151,0:04:04.195
and my uniform scales drawn on[br]the two axes, and let's plot

0:04:04.195,0:04:07.902
some points. Here's the first[br]Point, Point A and let's suppose

0:04:07.902,0:04:11.609
point a has coordinates 3. Two.[br]Let's see where that is.

0:04:12.890,0:04:17.916
Well, a has an X coordinate of[br]three, so we move in the X

0:04:17.916,0:04:19.352
Direction a distance 3.

0:04:21.510,0:04:25.267
And AY coordinate of two, so[br]we're moving up from the X axis

0:04:25.267,0:04:28.446
of distance to and that takes us[br]to this point here.

0:04:31.020,0:04:34.180
X coordinate of three Y[br]coordinate of two.

0:04:35.690,0:04:38.194
Let's have another point.[br]Let's have point B.

0:04:39.330,0:04:42.674
Switch those Round 2 three.[br]Let's see where that is now

0:04:42.674,0:04:46.322
with an X coordinate of two.[br]So we've got to move a

0:04:46.322,0:04:48.146
distance to from the Y axis.

0:04:49.600,0:04:53.040
And AY coordinate of three[br]means we move vertically. So

0:04:53.040,0:04:56.824
two across three up takes us[br]to this point there. And

0:04:56.824,0:04:58.200
that's my point, B.

0:05:00.000,0:05:03.124
Let's have some more. What[br]about this Point C, which has

0:05:03.124,0:05:06.532
an X coordinate of zero and a[br]Y coordinate of three? Let's

0:05:06.532,0:05:07.668
see where that is.

0:05:08.770,0:05:12.910
When X coordinate of 0 means we[br]move no distance at all in the X

0:05:12.910,0:05:14.842
direction, so we stay on the Y

0:05:14.842,0:05:19.924
axis. And we want to be a[br]distance three upwards from the

0:05:19.924,0:05:22.108
origin. So there's my Point C.

0:05:25.840,0:05:29.426
Point D that suppose that's[br]minus 3 zero. So I've introduced

0:05:29.426,0:05:33.338
a negative X cord in it. Now[br]remember with a negative X

0:05:33.338,0:05:37.576
coordinate, we're moving to the[br]left hand side of the Y axis. We

0:05:37.576,0:05:41.488
want to move a distance 3123, so[br]we're on this line here.

0:05:43.440,0:05:47.835
And AY coordinate is 0 means we[br]don't move up or down at all. So

0:05:47.835,0:05:49.593
at that point there. So there's

0:05:49.593,0:05:56.196
my point, D. Have a couple more[br]E minus 2 - 3 so we've got a

0:05:56.196,0:05:58.800
both a negative X and a negative

0:05:58.800,0:06:04.000
Y value. Minus 2 on the X[br]wings means we moved to the

0:06:04.000,0:06:08.480
left distance 2 - 3 on the Y[br]means we move down a distance

0:06:08.480,0:06:11.680
three, so we're down there to[br]get to point E.

0:06:13.750,0:06:18.200
Final one 2 - 2. Let's see[br]where that is.

0:06:19.410,0:06:23.034
X coordinate X coordinate of two[br]means. We moved to the writer.

0:06:23.034,0:06:26.435
Distance to. Y[br]coordinate of minus 2

0:06:26.435,0:06:29.419
means we moved down a[br]distance too, so

0:06:29.419,0:06:30.538
there's point F.

0:06:31.880,0:06:34.993
OK, so now given any ordered[br]pair of numbers representing a

0:06:34.993,0:06:38.672
point in the plane, you know[br]how to plot where it is. We

0:06:38.672,0:06:40.087
know how to locate it.

0:06:47.730,0:06:51.930
So now we know how to plot the[br]points. Let's see if we can plot

0:06:51.930,0:06:54.730
two points and find the distance[br]between the two points.

0:07:01.810,0:07:06.706
So some more graph paper here.[br]And let's suppose we pick any

0:07:06.706,0:07:11.602
two points and let me suppose[br]the first point, let's call it

0:07:11.602,0:07:13.234
a, has coordinates 13.

0:07:15.510,0:07:16.800
Let's put that on the graph.

0:07:17.980,0:07:21.230
X coordinate if one Y[br]coordinates of three means we

0:07:21.230,0:07:25.455
are one across and three up, so[br]at that point there that's a.

0:07:26.840,0:07:31.140
And let's choose my second[br]point to be four 5X

0:07:31.140,0:07:35.010
coordinate of four Y[br]coordinate of five puts me

0:07:35.010,0:07:35.440
there.

0:07:37.640,0:07:38.678
So two points.

0:07:39.800,0:07:40.700
Let's join them.

0:07:44.050,0:07:46.714
And that gives me my straight[br]line segment and what I'm going

0:07:46.714,0:07:48.490
to do is I'm going to show you

0:07:48.490,0:07:52.770
how we can calculate. The length[br]of this line segment, or In

0:07:52.770,0:07:55.150
other words, the distance[br]between points A&B.

0:07:56.620,0:08:00.780
Now, the way we do this is[br]as follows. We look at the

0:08:00.780,0:08:04.300
point A and will draw in a[br]horizontal line through a.

0:08:08.250,0:08:10.794
So because this line is[br]horizontal, it's parallel

0:08:10.794,0:08:12.066
to this X axis.

0:08:13.280,0:08:16.250
I'm going to draw in a[br]vertical line through be.

0:08:21.020,0:08:24.250
And again, because that's a[br]vertical line, it's parallel to

0:08:24.250,0:08:29.095
the Y axis, and so this angle in[br]here must be 90 degrees, and we

0:08:29.095,0:08:32.648
see what we've done is. We've[br]formed a right angle triangle.

0:08:34.870,0:08:37.918
Anywhere along this[br]horizontal line has a Y

0:08:37.918,0:08:40.966
coordinate of three, so in[br]particular the Y

0:08:40.966,0:08:44.014
coordinates at this point[br]here, where the right

0:08:44.014,0:08:45.919
angle is must be 3.

0:08:47.340,0:08:50.870
Call this Point C. It's Y[br]coordinate must be 3.

0:08:52.550,0:08:54.070
Now anywhere on this line.

0:08:55.380,0:08:59.730
Has an X coordinate of four the[br]same as the Point B, so the X

0:08:59.730,0:09:03.210
coordinate here must be 4. So[br]now we know the coordinates of

0:09:03.210,0:09:04.370
this point as well.

0:09:05.670,0:09:11.778
Let's put the coordinates of[br]being 45 and a in 1 three.

0:09:13.820,0:09:17.135
Now what I'm going to do is I'm[br]going to show you how we can

0:09:17.135,0:09:18.240
calculate this distance in here.

0:09:19.990,0:09:23.972
And this distance in here,[br]because then we will know the

0:09:23.972,0:09:27.954
lengths of two of the sides of[br]this right angle triangle.

0:09:29.490,0:09:32.310
And then we're going to use[br]Pythagoras theorem to get

0:09:32.310,0:09:34.848
this length, because this[br]length is the hypotenuse of

0:09:34.848,0:09:35.976
this right angle triangle.

0:09:38.070,0:09:39.438
Let's look at this[br]distance here.

0:09:41.190,0:09:44.682
Now the total distance from the[br]X axis up to point B.

0:09:45.710,0:09:48.550
Is 5. The Y coordinate of be.

0:09:49.480,0:09:52.860
So the distance all the way up[br]there is 5.

0:09:54.160,0:09:57.570
The distance from the X axis up[br]to Point C.

0:09:58.300,0:10:00.040
Is 3 the white coat?

0:10:00.110,0:10:03.197
You need to see so[br]that distance is 3.

0:10:04.430,0:10:07.346
So this distance that we're[br]really interested in here

0:10:07.346,0:10:10.262
must be the larger distance.[br]Subtract the smaller distance

0:10:10.262,0:10:11.234
5, subtract 3.

0:10:13.230,0:10:13.959
Which is 2.

0:10:15.380,0:10:18.084
It's important that you[br]recognize that we've calculated

0:10:18.084,0:10:21.464
this distance by finding the[br]difference of the Y coordinates,

0:10:21.464,0:10:22.478
5, subtract 3.

0:10:24.220,0:10:25.930
What about this distance[br]in here?

0:10:27.600,0:10:30.900
Well, the total distance[br]from the Y axis to this

0:10:30.900,0:10:32.220
dotted line is full.

0:10:33.390,0:10:36.679
Which is the X coordinate of be[br]or the X coordinate of C?

0:10:39.330,0:10:43.518
This shorter distance from[br]the Y axis to Point A is just

0:10:43.518,0:10:47.706
one. That's The X coordinate[br]of a, so that's one. So the

0:10:47.706,0:10:50.498
distance we're really[br]interested in is the larger

0:10:50.498,0:10:52.592
minus the smaller 4 - 1.

0:10:54.080,0:10:55.568
And 4 - 1 is 3.

0:10:56.520,0:11:00.095
And again 4 - 1 is the[br]difference of the two

0:11:00.095,0:11:03.020
X coordinates, 4 - 1,[br]giving you the three.

0:11:05.490,0:11:09.285
So now we know the length of the[br]base of this triangle is 3. We

0:11:09.285,0:11:10.550
know its height is too.

0:11:11.510,0:11:15.000
And we can use Pythagoras[br]theorem to find a B.

0:11:16.390,0:11:19.750
Pythagoras theorem will say[br]that AB squared.

0:11:21.390,0:11:26.780
Is the sum of the squares of the[br]other two sides. AB squared will

0:11:26.780,0:11:28.705
be 3 squared +2 squared.

0:11:30.910,0:11:35.950
3 squared is 9. Two squared is[br]four and nine and four is 30.

0:11:37.150,0:11:41.854
So AB squared is 13, so AB,[br]which is the distance we

0:11:41.854,0:11:46.166
want is going to be just the[br]square root of 13.

0:11:47.660,0:11:50.279
OK, so knowing the coordinates[br]of the two points.

0:11:51.110,0:11:55.036
We can find the lengths of the[br]sides of this triangle and then

0:11:55.036,0:11:58.056
use Pythagoras theorem to get[br]the hypotenuse, which is the

0:11:58.056,0:11:59.566
distance between the two points.

0:12:08.340,0:12:11.733
Now that's all well and good for[br]the specific case we looked at,

0:12:11.733,0:12:15.387
but what I want to do now is do[br]it more generally, instead of

0:12:15.387,0:12:18.258
having specific points, I want[br]to take two arbitrary points in

0:12:18.258,0:12:21.390
the plane and see how we can[br]find the distance between the

0:12:21.390,0:12:25.044
two of them. So again, let me[br]have a Y axis and X axis.

0:12:28.900,0:12:29.698
And and origin.

0:12:30.910,0:12:35.158
And let's pick our first point,[br]let's call it A and let's

0:12:35.158,0:12:39.052
suppose it's got an arbitrary X,[br]an arbitrary Y coordinate. So

0:12:39.052,0:12:43.654
let's call those X, one and Y[br]one. So there's my point A.

0:12:45.260,0:12:47.933
I'm going to take a second[br]point. Let's call that B.

0:12:50.210,0:12:53.324
And let's suppose its X[br]coordinate is X2, Y

0:12:53.324,0:12:56.438
coordinate is. Why two? So[br]these are arbitrary points.

0:12:56.438,0:12:57.822
Now could be anywhere.

0:13:00.250,0:13:03.671
We're going to try to find the[br]distance between the two

0:13:03.671,0:13:07.092
points, or In other words, the[br]length of this line segment.

0:13:09.020,0:13:11.740
As before, let's put[br]in a horizontal line.

0:13:13.180,0:13:13.930
Through a.

0:13:15.700,0:13:17.758
And a vertical line through be.

0:13:23.190,0:13:24.870
To form this right angle

0:13:24.870,0:13:27.826
triangle. Here, let's call[br]this Point C.

0:13:29.160,0:13:30.875
What are the coordinates[br]of points, see?

0:13:32.440,0:13:34.212
Well, anywhere along this

0:13:34.212,0:13:39.105
horizontal line. Will have a Y[br]coordinate, which is why one.

0:13:40.180,0:13:44.366
Because that's the same as the Y[br]coordinate of a. So the Y

0:13:44.366,0:13:46.620
coordinate of C must be why one?

0:13:48.110,0:13:51.949
What's its X coordinate? Well,[br]its X coordinate must be the

0:13:51.949,0:13:57.184
same as the X coordinate of bee[br]which is X2. So X2Y one are the

0:13:57.184,0:13:58.580
coordinates of Point C.

0:14:00.540,0:14:02.376
Let's calculate this[br]distance in here.

0:14:03.930,0:14:05.019
Now this distance.

0:14:05.590,0:14:09.110
Is the difference in the Y[br]coordinates the Y coordinates

0:14:09.110,0:14:14.038
of B subtract the Y coordinate[br]of C, so this is just Y 2

0:14:14.038,0:14:17.206
minus Y one. The difference in[br]the Y coordinates?

0:14:19.270,0:14:21.338
Similarly here this distance.

0:14:22.710,0:14:27.102
Is the X coordinate of C which[br]is X2 subtracted the X

0:14:27.102,0:14:31.860
coordinate of a which is X one.[br]So this distance is simply X2

0:14:31.860,0:14:32.958
minus X one.

0:14:34.320,0:14:38.954
And now you'll see, we know the[br]base and we know the height of

0:14:38.954,0:14:42.264
this right angle triangle and[br]we can use Pythagoras theorem

0:14:42.264,0:14:45.574
to get the hypotenuse, which is[br]the distance we require.

0:14:47.200,0:14:50.568
So Pythagoras theorem will say[br]that AB squared.

0:14:52.280,0:14:55.364
The square of the hypotenuse[br]is the sum of the squares of

0:14:55.364,0:14:56.392
the other two sides.

0:14:57.480,0:15:00.588
That's X2 minus X[br]one all squared.

0:15:01.890,0:15:05.950
Plus Y 2 minus[br]Y1 all squared.

0:15:08.110,0:15:14.670
And then if we want a B or we[br]have to do is square root both

0:15:14.670,0:15:20.000
sides, so maybe will be the[br]square root of X2 minus X one

0:15:20.000,0:15:23.280
all squared plus Y 2 minus Y one

0:15:23.280,0:15:28.642
or squared? And that is the[br]general formula that we can

0:15:28.642,0:15:32.314
always use to calculate the[br]distance between two arbitrary

0:15:32.314,0:15:33.946
points in the plane.

0:15:35.020,0:15:38.771
Let's look at a specific[br]example. Suppose we want to find

0:15:38.771,0:15:42.181
the distance between the points[br]AA, which has a coordinates

0:15:42.181,0:15:44.568
minus one and three and B which

0:15:44.568,0:15:48.475
has coordinates. Two and minus[br]four, for example, how can we

0:15:48.475,0:15:51.225
use this formula to find the[br]distance between these two

0:15:51.225,0:15:53.150
points without actually[br]drawing all the points?

0:15:54.570,0:15:58.720
Well, using the formula AB will[br]be the square root.

0:15:59.280,0:16:03.020
We want the distance, sorry. The[br]difference between the X

0:16:03.020,0:16:06.486
coordinates. So that's 2[br]minus minus one.

0:16:08.920,0:16:09.720
All squared.

0:16:11.620,0:16:14.970
Plus the difference[br]between the Y

0:16:14.970,0:16:17.686
coordinates, which is[br]minus 4 - 3.

0:16:19.950,0:16:20.700
All squared.

0:16:22.510,0:16:27.466
So that's the square root of 2.[br]Subtract minus one is 3 and 3

0:16:27.466,0:16:28.528
squared is 9.

0:16:29.670,0:16:35.410
Minus 4 - 3 is minus Seven and[br]minus 7 squared is plus 49.

0:16:36.840,0:16:40.860
So finally AB will be the square[br]root of 9 + 49.

0:16:42.310,0:16:43.159
Which is 58.

0:16:44.960,0:16:47.368
That's the distance between[br]those two points.

0:16:53.740,0:16:57.282
We're not going to look at how[br]you can calculate the

0:16:57.282,0:17:00.824
coordinates of the midpoint of a[br]straight line segment. Before we

0:17:00.824,0:17:02.434
do that, let me just.

0:17:03.210,0:17:06.486
Consider this problem. Suppose[br]we have an X axis.

0:17:07.280,0:17:15.022
And we have some points on the[br]X axis. So suppose we have one

0:17:15.022,0:17:17.234
2345, and so on.

0:17:18.330,0:17:22.158
Let's choose two points on this[br]X axis. Let's suppose we pick

0:17:22.158,0:17:25.990
this point. Where X is[br]2. Let's call that a.

0:17:27.170,0:17:32.162
And this point, where X is 4.[br]Let's call that be and ask

0:17:32.162,0:17:34.466
yourself what is the X value.

0:17:35.120,0:17:38.336
At the midpoint of the line[br]segment a be, so we're looking

0:17:38.336,0:17:42.356
at this segment Abe, and we want[br]to know what The X value is at

0:17:42.356,0:17:45.558
the midpoint. I think it's[br]fairly obvious from looking

0:17:45.558,0:17:48.462
at it that the X value at[br]the midpoint is 3.

0:17:49.510,0:17:52.543
Another way of thinking of that[br]is that 3.

0:17:53.170,0:17:55.438
Is the average of two and four.

0:17:56.370,0:18:00.213
Because if we average two[br]and four Adam together

0:18:00.213,0:18:05.764
divide by 2, two and four[br]is 6 over 2 which is 3.

0:18:07.220,0:18:10.650
So In other words, The X[br]coordinates at the midpoint

0:18:10.650,0:18:14.423
of a line segment is the[br]average of the X coordinates

0:18:14.423,0:18:15.795
at the two ends.

0:18:17.250,0:18:20.484
Just to convince you of[br]that, let's have a look at

0:18:20.484,0:18:21.072
another problem.

0:18:22.640,0:18:29.320
So we have one 23456[br]and Seven. And let's suppose

0:18:29.320,0:18:35.332
this time we pick, say,[br]the .3 for A.

0:18:36.080,0:18:38.248
And the point 74B.

0:18:39.080,0:18:42.626
And we're looking at this[br]straight line. Segments here

0:18:42.626,0:18:46.396
between A&B. Again, it's rather[br]obvious from looking at it.

0:18:47.810,0:18:51.170
The X value at the[br]midpoint is 5.

0:18:52.870,0:18:57.108
And if we average the X values[br]at the beginning and the end.

0:18:57.900,0:19:00.708
Of the line segment[br]three added to 7.

0:19:02.870,0:19:07.970
Divided by two as the average of[br]the X values is 10 over 2, which

0:19:07.970,0:19:12.863
is 5. So in a more general case,[br]in a few minutes time will see

0:19:12.863,0:19:16.126
that when we want to find the[br]coordinates of the midpoint of a

0:19:16.126,0:19:18.887
line, we just need to average[br]the values at either end.

0:19:23.610,0:19:28.355
OK, let's look at a more general[br]one. Suppose we have two points.

0:19:28.355,0:19:31.275
Suppose A is the .23, B is the

0:19:31.275,0:19:33.310
point. 45

0:19:35.350,0:19:39.580
Let's plot those. And then we're[br]going to try to find the

0:19:39.580,0:19:42.100
coordinates of the midpoint of[br]the line joining them.

0:19:59.920,0:20:04.100
OK. So as the .23,[br]so here's Point A.

0:20:07.160,0:20:10.355
And Point B is 45, so be will be

0:20:10.355,0:20:12.838
up here. And about that.

0:20:15.400,0:20:16.950
And there's my line segment.

0:20:18.890,0:20:19.410
Maybe?

0:20:21.180,0:20:24.780
We are interested in the[br]point which is the midpoint

0:20:24.780,0:20:28.020
of this line, so let's put[br]that on there.

0:20:29.400,0:20:30.816
And let's call it point P.

0:20:31.680,0:20:34.353
And because it's the[br]midpoint of the line, we

0:20:34.353,0:20:37.620
know that this length must[br]be the same as this length.

0:20:38.680,0:20:41.704
AP Is the same length as PB.

0:20:42.930,0:20:44.925
Let's put some other[br]lines in here.

0:20:46.330,0:20:47.820
A horizontal line through a.

0:20:49.690,0:20:51.050
Vertical line through be.

0:20:56.070,0:20:59.470
Horizontal line through P and a[br]vertical line through P.

0:21:01.050,0:21:04.306
And let's label some of these[br]points that we've created C.

0:21:05.060,0:21:07.430
And D. And he.

0:21:09.740,0:21:13.546
Now let's look at the line[br]segment from A to E.

0:21:15.910,0:21:16.420
OK.

0:21:19.440,0:21:20.380
Sorry Peter.

0:21:21.890,0:21:22.410
Not step.

0:21:23.980,0:21:24.510
OK.

0:21:28.010,0:21:28.300
Yeah.

0:21:33.220,0:21:35.746
Now let's look at the triangle.

0:21:36.390,0:21:40.350
ACP. And the

0:21:40.350,0:21:43.069
triangle BD. Sorry, the[br]triangle.

0:21:44.530,0:21:45.010
Yeah.

0:21:50.760,0:21:54.379
Now let's look at[br]the triangle ACP.

0:21:55.450,0:21:57.630
And the triangle PD.

0:21:58.330,0:21:58.760
Be.

0:22:00.480,0:22:03.120
Both of these triangles are[br]right angle triangles

0:22:03.120,0:22:05.760
because they both have a[br]right angle here.

0:22:08.020,0:22:10.513
Both of the triangles have[br]the same hypotenuse, which

0:22:10.513,0:22:13.283
has the same length 'cause[br]they both have the same

0:22:13.283,0:22:14.391
length there in there.

0:22:15.500,0:22:17.280
Also, this angle in here.

0:22:18.020,0:22:21.155
Is the same as this angle in[br]here because their corresponding

0:22:21.155,0:22:26.020
angles? So in fact, because[br]we've got two angles and a

0:22:26.020,0:22:29.287
side in the first triangle[br]corresponding to two angles

0:22:29.287,0:22:32.917
and the side in the second[br]triangle, these two triangles

0:22:32.917,0:22:34.732
or what we call congruent.

0:22:36.480,0:22:39.488
Congruent triangles are[br]identical in every respect, and

0:22:39.488,0:22:44.000
in particular. What this means[br]is that this length from B down

0:22:44.000,0:22:48.212
to D. Most corresponding[br]with the same as this length

0:22:48.212,0:22:49.757
from P down to see.

0:22:51.870,0:22:57.369
So In other words, because these[br]two lengths are the same BD and

0:22:57.369,0:23:00.330
PC, which is the same as DE.

0:23:01.480,0:23:03.508
Then this point D.

0:23:04.320,0:23:07.543
Must be the midpoint of the line[br]from A to B.

0:23:09.120,0:23:13.938
So the Y coordinate of D[br]must be the average of

0:23:13.938,0:23:16.128
the Y coordinates at B&B.

0:23:17.220,0:23:21.822
Now the average of the Y[br]coordinates at E&B is 3 + 5

0:23:21.822,0:23:26.778
/ 2, which is 8 over 2 which[br]is 4. So the Y coordinates

0:23:26.778,0:23:28.548
at D must be 4.

0:23:30.320,0:23:31.730
What about the X coordinate?

0:23:32.760,0:23:35.088
If we look at this line from A

0:23:35.088,0:23:38.756
to C. Because the triangles are[br]congruent, then this distance

0:23:38.756,0:23:43.061
from A to see must be the same[br]as the distance from P to D,

0:23:43.061,0:23:46.218
which is the same as the[br]distance from C to E.

0:23:47.080,0:23:50.740
So In other words, because these[br]two distances are the same, see

0:23:50.740,0:23:52.570
must be the midpoint of AE.

0:23:53.520,0:23:55.634
And because C is the midpoint of

0:23:55.634,0:24:00.736
AE. It's X coordinate must be[br]the average of the X coordinates

0:24:00.736,0:24:06.884
A&E. Now the average of those X[br]coordinates is 2 + 4, which is 6

0:24:06.884,0:24:11.317
/ 2 which is 3. So the X[br]coordinates here must be 3.

0:24:12.430,0:24:16.854
So the X coordinate is 3. We[br]know the Y coordinate is 3. At

0:24:16.854,0:24:20.962
this point we decided that the Y[br]coordinate was four and we know

0:24:20.962,0:24:22.858
the X coordinate is also 4.

0:24:24.050,0:24:28.873
So now we can read off what the[br]coordinates of point PRP will

0:24:28.873,0:24:30.728
have an X Coordinate A3.

0:24:32.790,0:24:34.850
And AY coordinate of four.

0:24:37.420,0:24:39.750
Now that's very complicated. It[br]seems very complicated, but in

0:24:39.750,0:24:42.080
practice it's very simple. Let's[br]just look at these numbers

0:24:42.080,0:24:45.030
again. The X coordinate at the

0:24:45.030,0:24:50.394
midpoint. Which is 3 is just the[br]average of the two X coordinates

0:24:50.394,0:24:55.946
2 + 46 / 2 is 3, the Y[br]coordinate at the mid .4 is just

0:24:55.946,0:24:59.416
the average of the two Y[br]coordinates. Five and three,

0:24:59.416,0:25:01.845
which is 8 / 2 is 4.

0:25:02.640,0:25:09.552
So in general, if we have[br]a situation where we've got a

0:25:09.552,0:25:13.008
point AX1Y one and point BX2Y2.

0:25:13.850,0:25:18.470
The coordinates of the midpoint[br]of the line AV are the average

0:25:18.470,0:25:20.010
of the X coordinates.

0:25:23.480,0:25:25.419
And the average of the Y[br]coordinates.

0:25:29.490,0:25:31.596
Let's just look at a[br]simple example just to

0:25:31.596,0:25:32.298
hammer that home.

0:25:34.460,0:25:40.900
Suppose we have a point A with[br]coordinates 2 - 4 and be with

0:25:40.900,0:25:45.500
coordinates minus four and[br]three. So two points and were

0:25:45.500,0:25:50.560
interested in the coordinates of[br]the midpoint of the line AB.

0:25:51.840,0:25:53.308
Select this point BP.

0:25:54.810,0:25:59.133
The X coordinate will be the[br]average of the two X

0:25:59.133,0:26:03.063
coordinates, which is 2. Added 2[br]- 4 over 2.

0:26:04.670,0:26:08.354
And the Y coordinate will be the[br]average of the two Y

0:26:08.354,0:26:11.424
coordinates, which is minus four[br]added to 3 / 2.

0:26:12.130,0:26:16.390
And if we tie do that up, will[br]get 2 subtract 4, which is minus

0:26:16.390,0:26:18.378
2 over 2, which is minus one.

0:26:19.250,0:26:24.830
And minus 4 + 3 is minus 1 - 1[br]over 2 is minus 1/2.

0:26:26.030,0:26:30.606
So there is the coordinates[br]of the midpoint of the line

0:26:30.606,0:26:31.438
joining amb.