1
00:00:00,920 --> 00:00:02,480
NARRATOR: Exercise 1.

2
00:00:02,480 --> 00:00:07,520
If a point, P, with coordinates
(x, y) lies on the line AB

3
00:00:07,520 --> 00:00:12,600
where A has coordinates (x1, y1)
and B has coordinates (x2, y2)

4
00:00:12,600 --> 00:00:19,040
and the ratio of AP to PB is a to b
with a and b both positive,

5
00:00:19,040 --> 00:00:22,640
show that x, that is the x-coordinate
of the point P,

6
00:00:22,640 --> 00:00:28,600
is equal to b times x1 plus
a times x2 all divided by a plus b,

7
00:00:28,600 --> 00:00:31,840
and y, that is the y-coordinate
of the point P,

8
00:00:31,840 --> 00:00:38,480
is equal to b times y1 plus
a times y2 all divided by a plus b.

9
00:00:40,680 --> 00:00:43,440
Let's first consider the situation
geometrically.

10
00:00:43,440 --> 00:00:47,760
And so we have the line AB
with the point P on that line.

11
00:00:47,760 --> 00:00:52,080
And the point P divides the line
in the ratio A to B.

12
00:00:52,080 --> 00:00:58,000
So for example, we could say that
the distance from A to P is a

13
00:00:58,000 --> 00:01:01,080
and the distance from P to B is b.

14
00:01:01,080 --> 00:01:07,800
And so that means the total distance
from A to B is a plus b.

15
00:01:09,560 --> 00:01:12,960
Now if we consider the two triangles
that we have here,

16
00:01:12,960 --> 00:01:16,240
we note first that they are in fact
similar triangles,

17
00:01:16,240 --> 00:01:20,040
so APP-dash is similar to ABB-dash

18
00:01:20,040 --> 00:01:23,360
due to the fact that the two
triangles have the same angles.

19
00:01:23,360 --> 00:01:29,200
And we also note that we now have
a triangle with a hypotenuse of A

20
00:01:29,200 --> 00:01:32,880
that's similar to another triangle
with a hypotenuse a plus b.

21
00:01:32,880 --> 00:01:35,480
And so we can work out
some scale factors here.

22
00:01:35,480 --> 00:01:38,240
So if we were to look at
the scale factor

23
00:01:38,240 --> 00:01:40,720
going from the smaller triangle
to the larger triangle,

24
00:01:40,720 --> 00:01:45,360
we would be multiplying the side
lengths by a plus b all divided by a.

25
00:01:45,360 --> 00:01:49,200
And if we were to go from the larger
triangle to the smaller one,

26
00:01:49,200 --> 00:01:54,120
we would be multiplying the side
lengths by a divided by a plus b.

27
00:01:55,720 --> 00:01:57,640
And so using that idea, then,

28
00:01:57,640 --> 00:01:59,360
we see that if we look at

29
00:01:59,360 --> 00:02:00,920
the horizontal distances,

30
00:02:00,920 --> 00:02:02,720
so dealing with the x-coordinates,

31
00:02:02,720 --> 00:02:06,080
we see that A to P-dash would be

32
00:02:06,080 --> 00:02:09,360
equal to a divided by a plus b

33
00:02:09,360 --> 00:02:11,600
multiplied by A to B-dash.

34
00:02:12,840 --> 00:02:16,680
And we can use this now to create
our required expressions.

35
00:02:16,680 --> 00:02:22,000
So AP-dash is equal to
a divided by a plus b times AB-dash.

36
00:02:22,000 --> 00:02:24,560
And we know that AB-dash
is the horizontal distance

37
00:02:24,560 --> 00:02:26,440
from the point A to the point B

38
00:02:26,440 --> 00:02:28,440
so that's just the difference between

39
00:02:28,440 --> 00:02:30,840
the x-coordinates of
those two points,

40
00:02:30,840 --> 00:02:33,200
that is x2 minus x1.

41
00:02:33,200 --> 00:02:38,320
And tidying up, we get a single
fraction - ax squared minus ax...

42
00:02:38,320 --> 00:02:44,080
Sorry, ax2 minus ax1
all divided by a plus b.

43
00:02:46,160 --> 00:02:48,440
Now, we're not just
looking for that distance,

44
00:02:48,440 --> 00:02:51,360
we're in fact looking for
the x-coordinate of the point P.

45
00:02:51,360 --> 00:02:54,000
And we know that the x-coordinate
of the point P

46
00:02:54,000 --> 00:02:56,760
is just the x-coordinate
of the point A

47
00:02:56,760 --> 00:02:59,120
plus this horizontal distance
between A and P

48
00:02:59,120 --> 00:03:00,880
that we've just worked out above.

49
00:03:00,880 --> 00:03:04,600
And so we get this expression
and we can now

50
00:03:04,600 --> 00:03:06,480
create common denominators,

51
00:03:06,480 --> 00:03:08,400
put the fraction together to find

52
00:03:08,400 --> 00:03:10,400
that the x-coordinate of the point P

53
00:03:10,400 --> 00:03:13,600
is b times x1 plus a times x2

54
00:03:13,600 --> 00:03:15,720
all divided by a plus b

55
00:03:15,720 --> 00:03:17,520
as was required in the question.

56
00:03:18,800 --> 00:03:23,640
We can now use similar logic for
the y-coordinate of the point P.

57
00:03:23,640 --> 00:03:30,720
And we note that PP-dash, that is
the vertical distance from A to P,

58
00:03:30,720 --> 00:03:35,000
is equal to a divided by a plus b
times BB-dash,

59
00:03:35,000 --> 00:03:38,440
which is the vertical distance
from A to B.

60
00:03:38,440 --> 00:03:41,160
And we know that this vertical
distance from A to B

61
00:03:41,160 --> 00:03:43,120
is simply the difference
in the y-coordinates

62
00:03:43,120 --> 00:03:44,600
between those two points,

63
00:03:44,600 --> 00:03:46,840
so that is y2 minus y1,

64
00:03:46,840 --> 00:03:49,480
which simplifies as a single fraction

65
00:03:49,480 --> 00:03:55,680
to a times y2 minus a times y1
all divided by a plus b.

66
00:03:55,680 --> 00:03:58,760
Again, though, the y-coordinate
of the point P

67
00:03:58,760 --> 00:04:01,840
does not equal this expression
in itself.

68
00:04:01,840 --> 00:04:05,240
The y-coordinate of the point P
would be equivalent to

69
00:04:05,240 --> 00:04:07,200
the y-coordinate of the point A

70
00:04:07,200 --> 00:04:10,880
plus this new vertical distance
that we've established above.

71
00:04:10,880 --> 00:04:12,560
And again, creating

72
00:04:12,560 --> 00:04:14,480
a common denominator and putting

73
00:04:14,480 --> 00:04:16,520
those two fractions together, we find

74
00:04:16,520 --> 00:04:18,240
that the y-coordinate of the point P,

75
00:04:18,240 --> 00:04:20,080
that is y, is equal to

76
00:04:20,080 --> 00:04:23,240
b times y1 plus a times y2

77
00:04:23,240 --> 00:04:26,280
all divided by a plus b,

78
00:04:26,280 --> 00:04:28,280
as was stated in the question.

79
00:04:28,280 --> 00:04:29,880
And so we have shown that

80
00:04:29,880 --> 00:04:31,640
the coordinates of the point P

81
00:04:31,640 --> 00:04:33,480
are as stated.