1
00:00:00,000 --> 00:00:00,320


2
00:00:00,320 --> 00:00:02,889
In this video, we're going to learn how to take the distance

3
00:00:02,890 --> 00:00:08,640
between any two points in our x, y coordinate plane, and

4
00:00:08,640 --> 00:00:11,060
we're going to see, it's really just an application of

5
00:00:11,060 --> 00:00:12,780
the Pythagorean theorem.

6
00:00:12,779 --> 00:00:14,039
So let's start with an example.

7
00:00:14,039 --> 00:00:16,070
Let's say I have the point, I'll do it in a darker color

8
00:00:16,070 --> 00:00:17,949
so we can see it on the graph paper.

9
00:00:17,949 --> 00:00:22,070
Let's say I have the point 3 comma negative 4.

10
00:00:22,070 --> 00:00:25,179
So if I were to graph it, I'd go 1, 2, 3, and

11
00:00:25,179 --> 00:00:26,480
then I'd go down 4.

12
00:00:26,480 --> 00:00:33,439
1, 2, 3, 4, right there, is 3 comma negative 4.

13
00:00:33,439 --> 00:00:40,109
And let's say I also have the point 6 comma 0.

14
00:00:40,109 --> 00:00:45,659
So 1, 2, 3, 4, 5, 6, and then there's no movement in the

15
00:00:45,659 --> 00:00:46,309
y-direction.

16
00:00:46,310 --> 00:00:47,609
We're just sitting on the x-axis.

17
00:00:47,609 --> 00:00:52,085
The y-coordinate is 0, so that's 6 comma 0.

18
00:00:52,085 --> 00:00:55,810
And what I want to figure out is the distance between these

19
00:00:55,810 --> 00:00:56,410
two points.

20
00:00:56,409 --> 00:01:00,349
How far is this blue point away from this orange point?

21
00:01:00,350 --> 00:01:01,910
And at first, you're like, gee, Sal, I don't think I've

22
00:01:01,909 --> 00:01:03,569
ever seen anything about how to solve for a

23
00:01:03,570 --> 00:01:04,439
distance like this.

24
00:01:04,439 --> 00:01:06,289
And what are you even talking about the Pythagorean theorem?

25
00:01:06,290 --> 00:01:08,180
I don't see a triangle there!

26
00:01:08,180 --> 00:01:09,690
And if you don't see a triangle, let

27
00:01:09,689 --> 00:01:10,939
me draw it for you.

28
00:01:10,939 --> 00:01:13,769


29
00:01:13,769 --> 00:01:18,060
Let me draw this triangle right there, just like that.

30
00:01:18,060 --> 00:01:20,689
Let me actually do several colors here, just to really

31
00:01:20,689 --> 00:01:23,090
hit the point home.

32
00:01:23,090 --> 00:01:24,650
So there is our triangle.

33
00:01:24,650 --> 00:01:25,990
And you might immediately recognize

34
00:01:25,989 --> 00:01:27,949
this is a right triangle.

35
00:01:27,950 --> 00:01:29,810
This is a right angle right there.

36
00:01:29,810 --> 00:01:33,519
The base goes straight left to right, the right side goes

37
00:01:33,519 --> 00:01:36,709
straight up and down, so we're dealing with a right triangle.

38
00:01:36,709 --> 00:01:39,209
So if we could just figure out what the base length is and

39
00:01:39,209 --> 00:01:42,349
what this height is, we could use the Pythagorean theorem to

40
00:01:42,349 --> 00:01:47,689
figure out this long side, the side that is opposite the

41
00:01:47,689 --> 00:01:49,700
right angle, the hypotenuse.

42
00:01:49,700 --> 00:01:55,890
This right here, the distance is the hypotenuse of this

43
00:01:55,890 --> 00:01:57,180
right triangle.

44
00:01:57,180 --> 00:01:58,180
Let me write that down.

45
00:01:58,180 --> 00:02:04,050
The distance is equal to the hypotenuse

46
00:02:04,049 --> 00:02:05,659
of this right triangle.

47
00:02:05,659 --> 00:02:07,479
So let me draw it a little bit bigger.

48
00:02:07,480 --> 00:02:10,710
So this is the hypotenuse right there.

49
00:02:10,710 --> 00:02:14,879
And then we have the side on the right, the side that goes

50
00:02:14,879 --> 00:02:16,069
straight up and down.

51
00:02:16,069 --> 00:02:18,500
And then we have our base.

52
00:02:18,500 --> 00:02:19,860
Now, how do we figure out-- let's

53
00:02:19,860 --> 00:02:22,550
call this d for distance.

54
00:02:22,550 --> 00:02:24,540
That's the length of our hypotenuse.

55
00:02:24,539 --> 00:02:27,639
How do we figure out the lengths of this up and down

56
00:02:27,639 --> 00:02:30,269
side and the base side right here?

57
00:02:30,270 --> 00:02:32,939
So let's look at the base first. What is this distance?

58
00:02:32,939 --> 00:02:35,009
You could even count it on this graph paper, but here,

59
00:02:35,009 --> 00:02:38,149
where x is equal to-- let me do it in the green.

60
00:02:38,150 --> 00:02:41,530
Here, we're at x is equal to 3 and here we're at x is equal

61
00:02:41,530 --> 00:02:42,800
to 6, right?

62
00:02:42,800 --> 00:02:44,550
We're just moving straight right.

63
00:02:44,550 --> 00:02:47,570
This is the same distance as that distance right there.

64
00:02:47,569 --> 00:02:49,909
So to figure out that distance, it's literally the

65
00:02:49,909 --> 00:02:50,710
end x point.

66
00:02:50,710 --> 00:02:52,240
And you could actually go either way, because you're

67
00:02:52,240 --> 00:02:53,770
going to square everything, so it doesn't matter if you get

68
00:02:53,770 --> 00:02:59,230
negative numbers, so the distance here is going to be 6

69
00:02:59,229 --> 00:03:03,359
minus 3, right?

70
00:03:03,360 --> 00:03:04,680
6 minus 3.

71
00:03:04,680 --> 00:03:08,520
That's this distance right here, which is equal to 3.

72
00:03:08,520 --> 00:03:10,140
So we figured out the base.

73
00:03:10,139 --> 00:03:12,609
And to just remind ourselves, that is equal to

74
00:03:12,610 --> 00:03:14,200
the change in x.

75
00:03:14,199 --> 00:03:17,099
That was equal to your finishing x minus your

76
00:03:17,099 --> 00:03:17,889
starting x.

77
00:03:17,889 --> 00:03:20,019
6 minus 3.

78
00:03:20,020 --> 00:03:21,860
This is our delta x.

79
00:03:21,860 --> 00:03:25,990
Now, by the same exact line of reasoning, this height right

80
00:03:25,990 --> 00:03:29,640
here is going to be your change in y.

81
00:03:29,639 --> 00:03:31,289
Up here, you're at y is equal to 0.

82
00:03:31,289 --> 00:03:32,669
That's kind of where you finish.

83
00:03:32,669 --> 00:03:34,359
That's your higher y point.

84
00:03:34,360 --> 00:03:37,900
And over here, you're at y is equal to negative 4.

85
00:03:37,900 --> 00:03:42,460
So change in y is equal to 0 minus negative 4.

86
00:03:42,460 --> 00:03:44,930
I'm just taking the larger y-value minus the smaller

87
00:03:44,930 --> 00:03:48,810
y-value, the larger x-value minus the smaller x-value.

88
00:03:48,810 --> 00:03:49,870
But you're going to see we're going to square it in a

89
00:03:49,870 --> 00:03:51,830
second, so even if you did it the other way around, you'd

90
00:03:51,830 --> 00:03:53,410
get a negative number, but you'd still get the same

91
00:03:53,409 --> 00:03:56,030
answer, so this is equal to 4.

92
00:03:56,030 --> 00:03:57,890
So this side is equal to 4.

93
00:03:57,889 --> 00:04:00,369
You can even count it on the graph paper if you like.

94
00:04:00,370 --> 00:04:02,310
And this side is equal to 3.

95
00:04:02,310 --> 00:04:07,039
And now we can do the Pythagorean theorem.

96
00:04:07,039 --> 00:04:11,729
This distance is the distance squared.

97
00:04:11,729 --> 00:04:12,399
Be careful.

98
00:04:12,400 --> 00:04:15,270
The distance squared is going to be equal to this delta x

99
00:04:15,270 --> 00:04:19,649
squared, the change in x squared plus

100
00:04:19,649 --> 00:04:23,439
the change in y squared.

101
00:04:23,439 --> 00:04:24,620
This is nothing fancy.

102
00:04:24,620 --> 00:04:26,769
Sometimes people will call this the distance formula.

103
00:04:26,769 --> 00:04:29,379
It's just the Pythagorean theorem.

104
00:04:29,379 --> 00:04:31,659
This side squared plus that side squared is equal to

105
00:04:31,660 --> 00:04:34,939
hypotenuse squared, because this is a right triangle.

106
00:04:34,939 --> 00:04:38,259
So let's apply it with these numbers, the numbers that we

107
00:04:38,259 --> 00:04:39,409
have at hand.

108
00:04:39,410 --> 00:04:43,020
So the distance squared is going to be equal to delta x

109
00:04:43,019 --> 00:04:47,579
squared is 3 squared plus delta y squared plus 4

110
00:04:47,579 --> 00:04:53,079
squared, which is equal to 9 plus 16, which is equal to 25.

111
00:04:53,079 --> 00:04:56,050
So the distance is equal to-- let me write that-- d squared

112
00:04:56,050 --> 00:04:57,550
is equal to 25.

113
00:04:57,550 --> 00:05:01,600
d, our distance, is equal to-- you don't want to take the

114
00:05:01,600 --> 00:05:03,160
negative square root, because you can't have a negative

115
00:05:03,160 --> 00:05:05,370
distance, So it's only the principal root, the positive

116
00:05:05,370 --> 00:05:09,019
square root of 25, which is equal to 5.

117
00:05:09,019 --> 00:05:10,779
So this distance right here is 5.

118
00:05:10,779 --> 00:05:13,129
Or if we look at this distance right here, that was the

119
00:05:13,129 --> 00:05:13,920
original problem.

120
00:05:13,920 --> 00:05:16,170
How far is this point from that point?

121
00:05:16,170 --> 00:05:20,180
It is 5 units away.

122
00:05:20,180 --> 00:05:22,930
So what you'll see here, they call it the distance formula,

123
00:05:22,930 --> 00:05:24,730
but it's just the Pythagorean theorem.

124
00:05:24,730 --> 00:05:27,000
And just so you're exposed to all of the ways that you'll

125
00:05:27,000 --> 00:05:29,769
see the distance formula, sometimes people will say, oh,

126
00:05:29,769 --> 00:05:33,339
if I have two points, if I have one point, let's call it

127
00:05:33,339 --> 00:05:37,579
x1 and y1, so that's just a particular point.

128
00:05:37,579 --> 00:05:42,879
And let's say I have another point that is x2 comma y2.

129
00:05:42,879 --> 00:05:45,759
Sometimes, you'll see this formula, that the distance--

130
00:05:45,759 --> 00:05:46,939
you'll see it in different ways.

131
00:05:46,939 --> 00:05:49,850
But you'll see that the distance is equal to-- and it

132
00:05:49,850 --> 00:05:52,170
looks as though there's this really complicated formula,

133
00:05:52,170 --> 00:05:53,930
but I want you to see that this is really just the

134
00:05:53,930 --> 00:05:55,370
Pythagorean theorem.

135
00:05:55,370 --> 00:06:05,340
You see that the distance is equal to x2 minus x1 minus x1

136
00:06:05,339 --> 00:06:18,649
squared plus y2 minus y1 squared.

137
00:06:18,649 --> 00:06:20,949
You'll see this written in a lot of textbooks as the

138
00:06:20,949 --> 00:06:22,199
distance formula.

139
00:06:22,199 --> 00:06:24,854


140
00:06:24,855 --> 00:06:27,250
And it's a complete waste of your time to memorize it

141
00:06:27,250 --> 00:06:30,060
because it's really just the Pythagorean theorem.

142
00:06:30,060 --> 00:06:32,370
This is your change in x.

143
00:06:32,370 --> 00:06:34,709
And it really doesn't matter which x you pick to be first

144
00:06:34,709 --> 00:06:36,729
or second, because even if you get the negative of this

145
00:06:36,730 --> 00:06:39,470
value, when you square it, the negative disappears.

146
00:06:39,470 --> 00:06:42,900
This right here is your change in y.

147
00:06:42,899 --> 00:06:44,692
So it's just saying that the distance squared-- remember,

148
00:06:44,692 --> 00:06:48,709
if you square both sides of this equation, the radical

149
00:06:48,709 --> 00:06:51,009
will disappear and this will be the distance squared is

150
00:06:51,009 --> 00:06:54,899
equal to this expression squared, delta x squared,

151
00:06:54,899 --> 00:06:59,089
change in x-- delta means change in-- delta x squared

152
00:06:59,089 --> 00:07:01,539
plus delta y squared.

153
00:07:01,540 --> 00:07:02,189
I don't want to confuse you.

154
00:07:02,189 --> 00:07:04,170
Delta y just means change in y.

155
00:07:04,170 --> 00:07:06,379
I should have probably said that earlier in the video.

156
00:07:06,379 --> 00:07:09,290
But let's apply it to a couple more, and I'll just pick some

157
00:07:09,290 --> 00:07:10,030
points at random.

158
00:07:10,029 --> 00:07:14,849
Let's say I have the point, let's see, 1, 2, 3, 4, 5, 6.

159
00:07:14,850 --> 00:07:17,660
Negative 6 comma negative 4.

160
00:07:17,660 --> 00:07:22,370


161
00:07:22,370 --> 00:07:24,430
And let's say I want to find the distance between that and

162
00:07:24,430 --> 00:07:32,290
1 comma 1, 2, 3, 4, 5, 6, 7, and the point 1 comma 7, so I

163
00:07:32,290 --> 00:07:34,370
want to find this distance right here.

164
00:07:34,370 --> 00:07:36,420
So it's the exact same idea.

165
00:07:36,420 --> 00:07:38,610
We just use the Pythagorean theorem.

166
00:07:38,610 --> 00:07:41,830
You figure out this distance, which is our change in x, this

167
00:07:41,829 --> 00:07:43,789
distance, which is our change in y.

168
00:07:43,790 --> 00:07:45,955
This distance squared plus this distance squared is going

169
00:07:45,954 --> 00:07:48,750
to equal that distance squared.

170
00:07:48,750 --> 00:07:50,600
So let's do it.

171
00:07:50,600 --> 00:07:53,160
So our change in x, you just take-- you

172
00:07:53,160 --> 00:07:54,130
know, it doesn't matter.

173
00:07:54,129 --> 00:07:56,170
In general, you want to take the larger x-value minus the

174
00:07:56,170 --> 00:07:58,400
smaller x-value, but you could do it either way.

175
00:07:58,399 --> 00:08:02,179
So we could write the distance squared is equal to-- what's

176
00:08:02,180 --> 00:08:04,430
our change in x?

177
00:08:04,430 --> 00:08:08,650
So let's take the larger x minus the smaller x, 1 minus

178
00:08:08,649 --> 00:08:09,429
negative 6.

179
00:08:09,430 --> 00:08:14,959
1 minus negative 6 squared plus the change in y.

180
00:08:14,959 --> 00:08:16,120
The larger y is here.

181
00:08:16,120 --> 00:08:17,530
It's 7.

182
00:08:17,529 --> 00:08:19,209
7 minus negative 4.

183
00:08:19,209 --> 00:08:22,549
7 minus negative 4 squared.

184
00:08:22,550 --> 00:08:24,430
And I just picked these numbers at random, so they're

185
00:08:24,430 --> 00:08:26,259
probably not going to come out too cleanly.

186
00:08:26,259 --> 00:08:30,490
So we get that the distance squared is equal to 1 minus

187
00:08:30,490 --> 00:08:31,280
negative 6.

188
00:08:31,279 --> 00:08:35,000
That is 7, 7 squared, and you'll even see it over here,

189
00:08:35,000 --> 00:08:36,168
if you count it.

190
00:08:36,168 --> 00:08:39,689
You go, 1, 2, 3, 4, 5, 6, 7.

191
00:08:39,690 --> 00:08:41,470
That's that number right here.

192
00:08:41,470 --> 00:08:43,399
That's what your change in x is.

193
00:08:43,399 --> 00:08:46,529
Plus 7 minus negative 4.

194
00:08:46,529 --> 00:08:47,720
That's 11.

195
00:08:47,720 --> 00:08:51,009
That's this distance right here, and you can count it on

196
00:08:51,009 --> 00:08:51,490
the blocks.

197
00:08:51,490 --> 00:08:52,649
We're going up 11.

198
00:08:52,649 --> 00:08:55,860
We're just taking 7 minus negative 4 to get

199
00:08:55,860 --> 00:08:57,740
a distance of 11.

200
00:08:57,740 --> 00:09:01,750
So plus 11 squared is equal to d squared.

201
00:09:01,750 --> 00:09:05,000
So let me just take the calculator out.

202
00:09:05,000 --> 00:09:13,265
So the distance if we take 7 squared plus 11 squared is

203
00:09:13,265 --> 00:09:16,169
equal to 170, that distance is going to be the square root of

204
00:09:16,169 --> 00:09:18,919
that, right? d squared is equal to 170.

205
00:09:18,919 --> 00:09:25,250
So let's take the square root of 170 and we get 13.0,

206
00:09:25,250 --> 00:09:28,250
roughly 13.04.

207
00:09:28,250 --> 00:09:30,549
So this distance right here that we tried to

208
00:09:30,549 --> 00:09:35,919
figure out is 13.04.

209
00:09:35,919 --> 00:09:38,539
Hopefully, you found that helpful.