For more information, please see full course syllabus of Math Analysis
For more information, please see full course syllabus of Math Analysis
Midpoints, Distance, the Pythagorean Theorem, & Slope
- To find the midpoint in one dimension, we take the average of the two numbers involved:
a+b
. - To find the midpoint in two dimensions (in the plane), we take the average location for each dimension on its own:
⎛
⎝x_{1} + x_{2}
, y_{1} + y_{2}
⎞
⎠. - To find the distance between two points in one dimension, we subtract one from the other. However, that could potentially cause a negative to pop up, so we deal with that by taking the absolute value of the result. Thus, the distance between any two
numbers is
|a−b|. - The Pythagorean theorem says, "On a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two legs." In other words, if the two legs (the shorter sides) are a and b, while the hypotenuse (the longest side) is c,
then we have
a^{2} + b^{2} = c^{2}. - The Pythagorean theorem allows us to find the distance between two points in the plane. We can plot the points, draw in a triangle, then figure out the lengths of the two legs. From there, we use the theorem to find the hypotenuse, which is the distance
between them. This gives the distance formula
d =
√
(x_{2} − x_{1})^{2} + (y_{2} − y_{1})^{2}
- Slope is a way to discuss how "steep" a line is. Another way to interpret it is the rate of change: the rate the line increases (or decreases) for every "step" to the right. We symbolize slope with m, and it is defined as any of the following
equivalent things:
m = rise
= vertical change
= y_{2} − y_{1}
. - Slope tells us how much the value of a line will change for every "step" to the right. A slope of m=−3 means that if we go 1 unit right, the line will drop down by 3 units. It is the line's rate of change.
- The idea of slope is very important in math (especially in calculus), so it's useful to have an intuitive sense of how slope works. Keep these facts in mind when thinking about slope:
- Positive (+) slope ⇒ line rises (when going right),
- Negative (−) slope ⇒ line falls (when going right),
- Bigger number (+ or −) ⇒ steeper line,
- m=1 ⇒ line rises at 45^{°} angle,
- m=0 ⇒ line is horizontal,
- m=−1 ⇒ line falls at 45^{°} angle.
Midpoints, Distance, the Pythagorean Theorem, & Slope
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
- Intro
- Introduction
- Midpoint: One Dimension
- Example of Something More Complex
- Use the Idea of a Middle
- Find the Midpoint of Arbitrary Values a and b
- How They're Equivalent
- Official Midpoint Formula
- Midpoint: Two Dimensions
- The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle
- Arbitrary Pair of Points Example
- Distance: One Dimension
- Absolute Value
- Distance: One Dimension, Formula
- Distance Between Arbitrary a and b
- Absolute Value Helps When the Distance is Negative
- Distance Formula
- The Pythagorean Theorem
- Distance: Two Dimensions
- Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem
- Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
- Slope
- Interpreting Slope
- Example 1
- Example 2
- Example 3
- Example 4
- Intro 0:00
- Introduction 0:07
- Midpoint: One Dimension 2:09
- Example of Something More Complex
- Use the Idea of a Middle
- Find the Midpoint of Arbitrary Values a and b
- How They're Equivalent
- Official Midpoint Formula
- Midpoint: Two Dimensions 6:19
- The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle
- Arbitrary Pair of Points Example
- Distance: One Dimension 9:26
- Absolute Value 10:54
- Idea of Forcing Positive
- Distance: One Dimension, Formula 11:47
- Distance Between Arbitrary a and b
- Absolute Value Helps When the Distance is Negative
- Distance Formula
- The Pythagorean Theorem 13:24
- a²+b²=c²
- Distance: Two Dimensions 14:59
- Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem
- Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
- Slope 19:30
- Slope is the Rate of Change
- m = rise over run
- Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
- Interpreting Slope 24:12
- Positive Slope and Negative Slope
- m=1, m=0, m=-1
- Example 1 28:25
- Example 2 31:42
- Example 3 36:40
- Example 4 42:48
Math Analysis Online
Transcription: Midpoints, Distance, the Pythagorean Theorem, & Slope
Hi; welcome back to Educator.com.0000
Today we are going to talk about midpoints, distance, the Pythagorean theorem, and slope.0002
We have a bunch of things to talk about.0006
The concepts in this lesson, like all the other introductory lessons, are all ideas you have seen in previous math classes.0008
None of this should be totally new to you, but we definitely want to review them.0013
There are some important concepts for the course in here.0017
We are going to be talking about these things later on; we are not going to directly talk about these other ideas.0020
We are not going to be teaching on them directly (other than this lesson, where we will be doing that directly).0025
But it is going to be assumed that you understand them all.0030
They are all going to repeatedly show up as we work on more complex things, so we really want to make sure0034
that all of these things are totally understood now, and that we really know what we are doing.0038
Not only that, but we really want to understand what we are doing.0042
We don't just want to be able to do these things; we want to understand how it works--how these formulas are operating:0045
not just how to use the formulas, but how they work--why they work--what they mean--what is causing them to be the way they are.0053
As we get into more advanced math, like this course right here, it is going to become more and more important0061
for you to understand the big picture--not just how you can do this one problem, but why doing the problem this way makes sense.0066
As we see more and more complex ideas, it is absolutely necessary for you to be able to make sense of why we are doing the things we are.0073
If you are just doing it because that is what you are told to, and that was the step that has to come next,0080
eventually things are going to fall apart, and you are not going to be able to see what the next step is going to be.0085
As you get older and older, you take more responsibility; as you get into more and more advanced subjects,0089
you are expected to understand what is going on and be able to take things on yourself.0093
Back in algebra, you were able to just take step-by-step formulas and apply them.0097
But now, you have to understand why those step-by-step formulas work,0101
because you have to understand why that works, so you can now tackle more complex ideas.0104
So, don't just understand how you can use these things, but understand what is going on on a deeper level.0109
That is what I really want you to get here, and what I want you to get out of the entire course at large.0113
That should be the goal of your education at this point: being able to understand what it is doing--0118
why it works--not just going through it so you can get the next grade.0123
All right, let's get started.0127
Let's say we want to find the point that is halfway between 0 and 4.0129
It seems pretty easy, right? Halfway between 0 and 4...well, there is 0; click over; click over; look, it is 2.0133
It is just half of 4: 4 over 2 equals 2.0140
Well, what if I want something a little bit more complex--like I want to figure out the midpoint between -5 and 17?0144
What point is halfway between those two?0149
There are two ways we could approach this idea.0151
Say we want to find the midpoint between -5 and 17: there are two ways to look at this question.0153
First, we can look at it through the idea of distance.0158
The distance between -5 and 17: how far would we have to travel to get from -5 to 17?0161
We would have to travel 22: - 5 to 0--we travel 5; 0 on to 17--we travel another 17.0166
We can also look at it from the point of view of 17 minus the one before it (-5); we get 22 either way we do it.0173
Technically, we haven't formally defined what distance means; we will in just a little bit.0180
But this makes sense: we can see that that should be 22.0184
So, to find the midpoint, what we can do is start at -5, and then we will work halfway up.0187
We will go up 22/2: we start at -5, and then we add 22/2.0194
22/2 is 11, so -5 + 11...we get 6: 6 is our answer; 6 is our midpoint.0199
Let's look at another way to do it, though: what if, instead, we wanted to find it through the idea of a middle?0206
We are looking for the midpoint, so it makes sense that the midpoint has to be halfway between them.0212
What is going to be halfway between them? Well, it would be the average of the two numbers.0217
What would be (if we could combine those two and figure it out) the most common place you have between the two?0221
What is going to be the middle between the two? It would be the average.0226
So, we take 17; we take -5; and we don't have to worry about the distance; we just realize, "Look, the midpoint is going to be halfway."0229
To be halfway, you have to be at the average of the two values.0236
So, we take -5 + 17, and we divide it by 2; that gets us 12/2, which is 6, the exact same thing we figured out last time.0239
That is great: it agrees with our previous work, and that is definitely what we want.0247
Since both of these things made logical sense to us, they had better both work; otherwise there is going to be a mistake somewhere in there.0250
All right, at this point, we could maybe look at this in general.0256
Let's say somebody hands us two points, A and B, and guarantees that A comes before B--that A is less than or equal to B.0260
We have this order that A will come before B, or maybe on top of B.0268
We don't know what they are, but we still want to be able to talk about the midpoint.0271
From our previous work, we have two ways to find this.0275
The distance from A to B is going to be B - A, so if we move up half of that distance, it is going to be (B - A)/2.0277
So, we start at A; and then we would add (B - A)/2; that is our half of the distance idea.0288
But we can also think, "Look, I know I am going to be here, and I know I am here; so I am looking for the place that is halfway between them."0293
So, we get the middle: that is (A + B)/2.0300
Now, from what we saw before...we saw both of these ways worked; as we had hoped, they give the same value.0304
We can go with either A + (B - A)/2 or (A + B)/2; they both are the same thing.0311
Let's prove that they are actually equivalent: if we start with our half-distance formula right here,0318
and we have A + (B - A)/2, well, let's try to put them on the same fraction.0323
We make it 2A/2; we can now combine fractions, and we get (2A + B - A)/2.0327
At this point, 2A - A...we get (A + B)/2; so sure enough, our half distance is equal to our middle; they are just two different ways of saying the same thing.0333
Since we have two ways to find the midpoint, and they are really just equivalent ways to get the same answer,0347
we will just, out of laziness, make one of them the official one.0352
A good motivator to do anything is because it is the easier way, as opposed to having to memorize two different things,0356
or go with the slightly longer one, let's go with the slightly shorter one; so we make our midpoint formula (A + B)/2; there we are.0361
Whatever the two points are that we are trying to find the middle between them, it is just (A + B)/2,0369
because we are just looking for the middle place, and the middle place must be the average of our two locations.0374
What if we wanted to do this in two dimensions, though?0380
What if we wanted to find the midpoint between (0,0) and (6,2).0382
Now, notice: we could look at this, as opposed to trying to figure out what is the midpoint on the line that connects the two of them0385
(I wish I had made that slightly more perfect)--instead of trying to figure out, "What is the middle going to be here?"0393
we can say, "Well, we know that there is going to be some distance vertical and some distance horizontal."0397
It must be that it splits our distance halfway horizontally, and it splits our distance halfway vertically.0406
So, the two of them come together, and that is our midpoint.0413
It is going to have to be the horizontal middle: our horizontal distance was 6, so 6/2 is where we are going.0419
Our vertical distance was 2, so it is 2/2, so our location (that is going to be halfway between them)0425
will be half of our horizontal distance and half of our vertical distance: (6/2,2/2), or (3,1).0431
The midpoint is going to occur at the horizontal middle and the vertical middle, put together into a single point.0439
What if we are doing this for some arbitrary pair of points, (x_{1},y_{1}) and (x_{2},y_{2})?0446
Well, the same basic idea: we can think, "What is the vertical, and what is the horizontal?"0450
What are going to be the midpoints of those two things?0457
We think with that idea, and we are able to come up with the same logic that is going to occur at the horizontal and vertical middles.0459
This point is going to be the same vertical height, because we never changed height as we went along.0466
So, horizontally, we are going to have changed to a new horizontal location; but vertically, this thing right here is going to be the same thing here.0473
The same sort of idea here: y_{2} is going to change when we switch down, as we go down.0484
But horizontally, we didn't change there; so we have fixed things here, so the point that we are meeting the two at--0488
if we drop a perpendicular and throw out a horizontal, we are going to meet up at (x_{2},y_{1}).0498
So, the midpoint horizontally is going to be (x_{2} + x_{1})/2,0505
which will get us the middle location, because it is the average of our two horizontal locations.0516
The average of our two vertical locations is going to be (y_{2} + y_{1})/2.0521
We bring these two things together, and we get where our middle is--we get our midpoint that way.0528
So, from the midpoint in one dimension, we can figure out what it is horizontally.0538
The horizontal motion was x_{1} + x_{2}, so its midpoint is (x_{1} + x_{2})/2.0542
And vertically, our locations were y_{1} and y_{2}, so the middle of our vertical locations will be (y_{1} + y_{2})/2; great.0548
Our midpoint formula is just (x_{1} + x_{2})/2, and (y_{1} + y_{2})/2; awesome.0559
The next idea: distance--what if we want to find the distance between 2 and 7.0567
That is easy: 7 minus 2 equals 5--done, right?0571
Well, we could make a mistake, though; we are not perfect; what if we accidentally put it in in the wrong order.0575
We put in 2 - 7 = -5: well, that doesn't really make sense, because distance has to be a positive length.0581
There is no such thing as a negative length, if we are measuring something; you can't say, "Oh, that man is -2 meters tall."0589
It doesn't make sense; we can't talk about his distance, his length, as being -2.0595
So, -5 doesn't really work; but notice, 5 and -5...they are very different in one way; but in another way, they are very similar.0599
One of them is the same thing, just with a negative sign; the other one is the same thing, but with a positive sign.0610
So, they are the same number, but with different signs on them.0616
In one way, we can think of 5 and -5 as being very different numbers; they are opposites, after all.0620
But in another way, we can think of them being the same number, but with different signs.0624
They are the same distance from 0; so what we really want is some way of being able to force "positive-ness."0628
5 and -5 are pretty close to both being the same thing; it is just that one of them is the wrong sign.0637
So, if we could force it to be positive, it wouldn't matter if we did 7 - 2 or 2 - 7,0643
because, since we are forcing positive, it will always give us the same thing.0648
We would always find that distance, even if we put it in the wrong way.0651
This is where the idea of absolute value comes in.0654
We call this idea of forcing a positive, making something always come out as positive, absolute value.0657
It is represented by vertical bars on either side.0664
So, whatever we want to take the absolute value of, we just put inside of two vertical bars.0666
We could have |x - 5|, and whatever comes out after we plug in x, we would take its absolute value--we would force positive-ness.0670
It is going to take negatives, and it will make them positive; and it will take positives and not do anything; also with 0, it won't do anything.0677
If you are positive, you stay positive; if you are 0, you stay 0; if you are negative, you flip to being the positive version.0683
You hit it with another negative: so -5 would become 5, but positive 5 would just become positive 5, as it already started.0689
-47 would become 47; 47 would just stay as 47; great.0697
With this idea of absolute value, we can now tackle how we talk about distance in one dimension.0703
So, if we want to talk about it just arbitrarily, if we have two points, A and B, and we don't know which comes first--0707
we don't know if it is going to be A then B, or B then A--we have no idea which comes first (if it is A first or B first)--0714
but we still want to be able to talk about what the distance is between them,0726
well, our previous logic can tell us that one of these two is going to be right: A - B or B - A.0729
But the other one is going to be wrong, although almost correct, because it will be the negative version.0735
So, we have A - B versus B - A: we want some way of being able to say, "Let's just get rid of the negative signs," right?0741
Let's force everything to be positive; then, it doesn't matter what order we put it in, because it is going to be the same distance,0750
because it is just a negative version or a positive version;0756
it doesn't matter, because we will flip everything to positive; we will always get the distance.0758
So, we toss some absolute values on there: absolute value to the rescue!0761
We wrap them in absolute values, and they both become the same positive, correct distance.0764
So, the absolute value of A - B is the same thing as the absolute value of B - A,0770
because the only difference would be whether it is negative or positive.0774
And now they are both forced to be positive, so |A - B| is equal to |B - A|,0776
which is just going to be the distance between A and B, which is the distance between B and A.0782
For ease, we will just make the first one official: so the absolute value of A - B is the distance between those two locations.0787
We just take the absolute value of the difference, and that gives us how far the two things are apart0794
when it is in one dimension--when we are just on the number line.0799
What if we are in more dimensions, though? Let's take a look at the Pythagorean theorem, because we will need that to discuss two dimensions.0803
To discuss distance in two dimensions, we need to understand the Pythagorean theorem.0809
You have probably learned this before; if it isn't really something that you know well, you are going to want to go back and relearn it.0813
Make sure you have this idea, because it is going to show up all sorts of places in precalculus and in calculus.0817
And it is definitely going to show up a whole bunch in the trigonometry portion of this course.0823
So, definitely make sure that you go and re-study it if you don't remember it.0826
What it was: we have a right triangle (a right angle in the corner): the square of the hypotenuse0829
(that is the long side, the side that is opposite our right angle) is equal to the sum of the squares of the other two legs.0835
So, we square each of the other two, smaller legs.0846
And when we add them together, a^{2} + b^{2}...each of the smaller legs squared,0849
then added together...that is going to be equal to our hypotenuse, squared.0854
a^{2} + b^{2} = c^{2}: leg 1 squared, plus leg 2 squared, equals hypotenuse squared.0859
That is the idea of the Pythagorean theorem.0866
So, any time we see a right triangle showing up, anything we have showing up with perpendiculars--0868
it is a good idea to think, "Oh, I wonder if I could use the Pythagorean theorem here."0873
It will be very, very useful in a whole bunch of situations.0877
If you are not really comfortable with using it at this point, definitely go back and review this idea.0880
Either search for it on the Internet, or just try to do a couple of exercises and make sure you have practiced on it.0883
Or go review it on Educator.com: listen to the lecture, and then practice some exercises.0888
But you want to make sure that you are definitely comfortable with the Pythagorean theorem,0891
because it is going to show up a whole lot for the rest of the time you are doing math.0894
All right, on to distance in two dimensions: what if we wanted to find the distance between (0,0) and (6,8)?0898
Well, we can't just subtract and take absolute values, because we have two dimensions that we are running in.0904
We have to deal with both of these things at once.0910
What we do is say, "Let's turn this into a triangle."0912
We drop a perpendicular from (6,8); we now have this right angle in the corner.0916
And with this right angle in the corner, we can use the Pythagorean theorem.0921
When we did midpoint, we dropped down perpendiculars; we drew out horizontals and perpendiculars.0926
And we were able to get a right triangle going on, which will help us to find middle locations for horizontal and vertical.0931
Now, we are allowing us to find horizontal lengths and vertical lengths.0937
We break it into horizontal and vertical parts.0940
So, if we are at (6,8) up here, then the distance that we traveled horizontally is 6.0943
The distance that we traveled vertically is 8: remember, 6 is because that is the horizontal portion; 8 is because that is the vertical portion.0949
So, we use the Pythagorean theorem: we know that d^{2}, the diagonal, the hypotenuse,0957
is going to be equal to 6^{2} + 8^{2} = 36 + 64 = 100.0961
So, for our diagonal, our distance, d, equals 10.0968
So, we can figure out that this has to be 10, up here on that side, because we can turn it into a right triangle, which allows us to apply the Pythagorean theorem.0972
What if we look at this in a more general way, where we just get two arbitrary points,0982
where we don't know what they are--(x_{1},y_{1}), our first point, and (x_{2},y_{2}).0985
Now, I didn't talk about this explicitly the first time, but when I say x_{1}, I am not saying x times 1; I am just saying our first x.0990
x the first, y the first, x the second, y the second--(x_{1},y_{1}),(x_{2},y_{2})--0996
that is what you should interpret when you see those little subscripts, those little numbers on the bottom right.1003
So, (x_{1},y_{1}), (x_{2},y_{2}): they are just two arbitrary points, sitting out in a plane.1008
We can continue with this idea: we will make a triangle.1013
We will toss out a horizontal from this one; we will go straight with a horizontal out.1016
And we will drop directly down with a vertical, like this; and that will guarantee us that we have a right triangle that we can now work with.1023
And now, we have a way of being able to talk about the distance of that.1034
So, we draw that in, and we can say, "Oh, what is the horizontal length?"1037
Well, since we ended up at x_{2} (because it is going to have the same horizontal location as our second point),1040
we went from x_{1} to x_{2}; our distance is the absolute value of (x_{2} - x_{1}).1047
The horizontal length is going to be the absolute value of (x_{2} - x_{1}).1053
What is the vertical length--what is the vertical leg of our triangle?1056
Well, we end up at y_{2}; and what is the location that we are starting on this triangle?1059
It is going to be y_{1}, because it is going to be the same as over here.1064
So, we take y_{2} - y_{1}; the absolute value of that is going to be our vertical length.1067
The length of the vertical leg of the triangle is the absolute value of (y_{2} - y_{1}).1074
So, if we want to know what the distance of the diagonal is, it is going to be d^{2} = (|x_{2} - x_{1}|)^{2} + (|y_{2} - y_{1}|)^{2}.1078
Now, there is a little thing that we can notice at this point.1096
OK, if I have |x_{2} - x_{1}|, and then I square it, well, if I just take1099
x_{2} - x_{1}, and I square that, that is going to be the same thing.1104
Remember, if I have (-7) squared, that comes out to be 49, which is the same thing as 7 squared.1108
So, we don't have to take an absolute value to begin with, because, when it is the number times itself, if it has a negative,1116
if it multiplies by itself with another negative, those negatives are going to cancel each other out.1123
But if we start on a positive, we are going to have no negatives anyway.1127
So, the absolute value of x_{2} - x_{1}, squared, is equal to the quantity1130
(x_{2} - x_{1}), squared, because they are both going to come out to be positive, in any case.1134
So, this is also the same for y; so we can actually drop our absolute values--we don't have to worry about absolute values when we are doing this.1139
And the distance will be equal to the square root, because it is d^{2} equals this thing squared, plus this thing squared1144
(horizontal length squared, plus vertical length squared); so d equals the square root--take the square root of both sides, so we get just d.1151
It is the square root of ((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}).1157
So, it is the difference in our two horizontal locations, squared, plus the difference in our two vertical locations, squared; great.1163
That is distance in two dimensions.1168
All right, slope: slope is a way to discuss how steep a line is--how quickly it is going up--how much it is changing one way or the other.1171
Another way to interpret is the rate of change--the rate that the line increases or decreases for every step to the right.1181
So, if we take one step over to the right, it tells us how much up we should go or how much down we should go,1190
depending on if it is a positive slope or a negative slope.1195
So, it is one step over, and we change by the slope.1198
Either way we look at slope--whether we look at it as how steep it is (the angle we are working at)1202
or if we look at it as the rate of change (how much we are changing for every step we take on the line), we define it the same way.1206
We take some arbitrary portion of the line--any chunk of the line--and we see how much it "rises" and how much it "runs."1213
So, rise is the vertical amount of change, and run is the horizontal amount of change.1219
So, if we had some chunk of line, like this, then what we would do is just set two arbitrary points, here and here.1224
And then, we would say, "OK, how much did we move vertically?" (that is our run) and "How much did we move horizontally?"...1233
Oops, sorry, not our run--I said the wrong thing there: our vertical change is our rise--you rise vertically.1243
And our horizontal change is our run, because you run along the ground; you run along (generally) horizontal things.1251
So, our rise, compared to our run--we divide the rise by the run.1258
We symbolize it with m: why do we symbolize it with m? Because, clearly, the first letter of slope starts with m, right?--it makes sense.1262
I am kidding; there is actually no good reason, and nobody knows why we use m.1270
Anyway, m equals rise over run; that is how we symbolize it--the amount that we rise by, divided by the amount that we run.1274
We can also talk about this, since rise and run are just other words for vertical change and horizontal change.1281
The slope is equal to the amount of vertical change in our line, divided by the amount of horizontal change in our line.1286
Now, keep in mind: vertical change could go down, at which point we would have a negative rise (I keep accidentally swapping them).1292
We would have a negative rise if we ended up dropping down.1300
All right, let's go back really quickly and address this asterisk.1305
Why is it that we can look at any arbitrary portion of the line?1308
Why doesn't it matter which section of the line we look at?1312
Shouldn't it matter if we look at a big section or a small section, or if we look at a high section or a low section?1315
No, because the line never changes slope: that is what it means to be a line.1320
Every section of it is going along at the same steepness; every section of it is going along at the same rate of change.1324
If we put a bunch of sections together, they will all agree on their slope.1331
If we look at just one tiny section in a very different place, it is still going to have the same slope.1334
Whatever part of the line we look at, it will always have the same slope.1338
So, we get the same value for slope, no matter where we look on the line.1341
So, that is why we don't have to worry about what portion of the line we are considering.1345
We just look somewhere, and that is our slope; great.1349
So, if we have two points, they can define a line; so we want to find the slope between two arbitrary points, (x_{1},y_{1}) and (x_{2},y_{2}).1353
Well, what we have to do is say, "All right, how much did I rise in that chunk?"1361
And I compare it to how much I ran in that chunk.1368
We figure out both of those, and we will be able to get what our slope has to be.1372
So, we build a right triangle to help us find these distances.1376
Since this here is going to be the same as the horizontal of our second point, that matches up there.1379
And it is going to be the same as the vertical, since vertical doesn't change as we go horizontally.1385
Those match up there, so the amount of run that we have is (x_{2} - x_{1}).1388
That is how much we changed as we went from left to right.1395
And the amount vertically that we changed is (y_{2} - y_{1}), because we went up from y_{1} to y_{2}).1398
We went up from y_{1} to some y_{2}; so it is y_{2} - y_{1}.1405
So, our rise is y_{2} - y_{1}, and our run is x_{2} - x_{1}.1411
Our slope is equal to the rise divided by the run, which is our vertical change divided by our horizontal change.1417
That gets us (y_{2} - y_{1})/(x_{2} - x_{1}).1426
We have been using this formula for years, but now...hopefully, you actually understood it before...1429
but even if you didn't understand it before, why this formula was slope,1434
hopefully now you are thinking, "Oh, now I see why slope is what it is!"1437
It is because it is just coming from rise over run; that is how we defined it; so we get (y_{2} - y_{1})/(x_{2} - x_{1}).1440
because (y_{2} - y_{1} is how much we rose, and (x_{2} - x_{1}) is how much we ran.1446
Being able to understand what we are doing with slope, though, requires being able to interpret it on an intuitive basis.1452
We want to know what to immediately imagine when we are talking about something that has a slope of 50.1457
So, slope tells us how much the value of a line changes for every step to the right.1464
If we have a slope of m = 2, then that means, if we take one step to the right, then we will go up 2 steps.1469
Our line will end up looking like this.1480
If we had a slope of, say, -3, then it would be that for one step to the right, we take 3 steps down, so our line would look like that.1487
What we have here is a way of being able to talk about the line's rate of change--how much you change for one click.1508
You click over, and you change by your slope.1514
So, we can think of slope as how steep it is (bigger numbers will make it steeper, because it means more steps to be made in our rate of change).1517
All right, there is one step, but how many times we go down or how many times we go up is the number of our slope.1524
So, a line's rate of change is its slope: it is a way of talking about how fast this line is changing as we slide along it.1530
You want to keep these facts in mind as we think about slope.1539
If we have a positive slope, it means that the line is rising; we are always thinking about it as we go from left to right.1543
That is how we are always reading how our slope works: it is always what happens as we go from left to right.1552
So, positive slope means we rise as we go to the right; a negative slope means that we fall when we go to the right.1558
We either go up by positive, or we go down because it is negative.1565
A bigger number, whether it is positive or it is negative, means a steeper line.1568
The steeper the line is, the bigger the slope has to be; the bigger the slope is, the steeper the line is.1573
A big slope, like, say, m = 50, is going to be really, really super steep.1581
It is going to go up really, really fast, because for every step it takes to the right, it has to take 50 steps up.1587
Similarly, in m = -50, it is going to be very similar; but for every step it takes to the right, it takes 50 steps down; so it is super steep going down.1592
A big number, whether it is a big positive number or a big negative number--that is going to imply a steep line.1603
Some specific locations to keep in mind: if m is equal to 1, then that means our line rises at a 45-degree angle,1609
because for every step to the right, we take one step up.1615
So, it means that we have a nice, even-sided triangle: 45...these two have to be the same.1618
If we have m = -1, then for every step we take over, we take a step down.1626
So, we have the same idea; but instead, we are going down now; so these two angles have to be the same.1633
We have a nice 45-degree angle in that triangle as well--what makes up the line.1639
So, we are either rising at 45 degrees (if we have a positive one) or we are falling at 45 degrees (if we have a negative one).1644
And if m = 0, then we take one step over; we take no steps up; we just continue taking steps over forever and ever.1650
So, m = 0 means our line is horizontal; m = 1 means our line rises at 45 degrees; m = -1 means the line falls at 45 degrees.1657
This also means that everything between positive 45 and -45 is all going to happen in fractions--things that are between -1 and 1.1664
If we want to get really steep lines, that is as we approach either positive infinity, or as we approach negative infinity.1676
We can never be perfectly vertical with a slope, because that will require either positive infinity or negative infinity.1681
And we are not able to actually call those out, because they are not really numbers.1689
But as we go from 1 and click up more and more and more, and approach infinity more and more and more,1692
we will need larger and larger numbers to become steeper and steeper and steeper.1696
All right, there are lots of ideas that we have covered here; now we are ready to start talking about some examples.1700
First, the idea of midpoints: if we have a midpoint, and we are looking between -3 and 37, then remember,1706
our formula for midpoints was just (A + B)/2; it is the average of the two.1712
So, the average of -3 and 37...put those two together: we get 34/2, which equals 17; so, 17 is our answer.1718
If we want to find (6,2) to (1,-12), then we do it on each of the components, because we look at the horizontal average, and we look at the vertical average.1731
So, (6 + 1)/2, the average of our horizontal components, and (2 + -12), the average of our vertical components, will get us 7/2 and -10/2.1743
7/2...we can't simplify that anymore, so that will lock in; but -10/2...we can simplify that, so we get -5.1761
(7/2,-5): that is our midpoint for this one right here.1769
And our last one: what if somebody handed us things that weren't numbers--they hand us 2a, 3b, 6k, -7b?1775
They are numbers, in that a represents some number--it is a placeholder--it is a variable.1782
b represents a number; k represents a number; they all represent numbers.1786
But we can't actually solve and get numbers, like we did with these previous two ideas, these previous two questions.1789
But we can still use the numbers--we can still use the variables.1796
We just put them into the formula, just the same: we are still looking for what is the average of our horizontals--what is the average of 2a and 6k.1799
What is the average of our horizontal locations?1808
And what is the average of our vertical locations, 3b + -7b?1811
We are still looking for the same sort of average ideas, horizontal and vertical.1817
It just is that we can't combine 2a and 6k, because a and k are speaking totally different languages.1822
And b and b, we can combine, because they are speaking the same language.1828
So, 2a + 6k--we can't combine that, but we can have our fraction, the denominator,1831
go onto both of them: 2a/2 + 6k/2; we have the denominator split onto both of them.1835
And 3b + -7b; that begins -4b/2; so 2a/2 becomes just a; 6k/2 becomes 3k; no comma--they are combined together through addition.1842
But they can't do anything more: a and k don't speak the same language, so they can't combine.1856
But we have a + 3k; we know that is what our horizontal location is.1861
So, if we were given a and k later, we can easily get what the midpoint is, in terms of actual numbers.1864
-4b/2: that is -2b; there we are--we don't have numbers in the terms of 53 or something,1869
but we have answers that are still pretty good.1879
If we get what these variables are later--if we somehow get them because we solve for them,1882
or somebody hands them to us--we will be able to immediately find out what actual numbers would be.1886
And this gives us a great idea of where the midpoint is, based on variables.1890
We don't have to be working with numbers to be able to solve for these things; we can also just put in variables,1894
and just follow the exact same rules that the numbers would follow.1898
The next one: let's talk about distance--what is the distance between -7 and 8?1903
Remember, we do this based off of the difference between the two numbers, its absolute value.1907
So, we could take |-7 - 8| or |8 - -7|; either way we do this, -7 - 8 will be -15; 8 - -7 will turn that into positive 15.1914
Either way, they both equal 15; so the answer is 15.1929
The distance between -7 and 8 is 15, which makes sense, because -7 clicks up to 0 by going 7, and 0 clicks up to 8 by going 8; so 7 + 8 is 15.1933
Great, that makes a lot of sense.1943
What if we want to figure out what the distance is between (3,7) and (9,-1)?1945
Well, remember: now we are working off of what we figured out before, with the Pythagorean theorem and how that applied to distance.1949
So, it is going to be the square root of the difference between our horizontals, squared, plus our difference between our verticals, squared.1955
It is the square root of all of those things.1965
So, formulaically, it is d = √[(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}]--the square root of all of that.1967
So, in this case, let's arbitrarily set this as our second one; and we will set this as our first one.1981
The distance is equal to the square root of (x_{2} - x_{1})^{2}...that would be...1987
the second x is 9, minus...the first x is 3, squared; plus our y portion: the second y is -1, minus the first y (is 7), squared.1995
So, distance is equal to the square root of 6 squared plus -8 squared; distance equals √(36 + 64),2009
which means distance is equal to √100, which equals 10.2024
So, the distance between those two points is 10.2028
The final one: what if we get some things that don't turn out nicely?2031
We have these ugly decimal numbers: we still follow the exact same idea.2035
The distance is equal to the square root of...arbitrarily, we will just make this one the first one, and this one the second one...2039
the answer would turn out the same, because of all of the things that we talked about before.2045
It doesn't matter which gets turned into the second and which gets turned into the first; the distance is going to be the same between them.2048
If that doesn't really make sense, go back to when we talked about how we figured out that this is the formula.2053
And notice that (x_{2} - x_{1})^{2} is the same as (x_{1} - x_{2})^{2}.2057
It comes up with the ideas where we were talking about absolute value before.2063
So, the first x is -0.2, minus the second x (is 2.5), squared, plus...the first y is 1, minus the second y (is 1.7), squared.2066
So, distance equals the square root of...-0.2 - 2.5 becomes -2.7, squared; plus...1 - 1.7 becomes -0.7, squared.2084
We use a calculator to figure this out; or we could do it by hand, but I used the calculator when I figured it out.2100
And I figured out before that (-2.7)^{2} becomes 7.29, plus...(-0.7)^{2} becomes 0.49,2105
because those negatives just end up canceling with each other when they square against themselves.2115
So, this is 7.29 + 0.49; so our distance is equal to the square root of 7.78 (what we get when we combine these two numbers--7.78).2119
That is the distance between those two things.2129
Of course, the square root of 7.78 is kind of hard to actually use, if we had to measure something and cut something,2132
if we were making, say, a bookshelf (who knows?); so we could approximate this.2139
We could take the square root of 7.78 with our calculator, and that would be approximately equal to 2.79.2144
In reality, the decimals actually keep going, and they keep shifting around forever, because 7.78 is not a perfect square.2149
So, we can't just come up with a nice, easy number; but 2.79...we can just cut it off, and we will say, "Yes, 2.79 is pretty good."2155
So, we get an approximate value; the true answer, though, is √7.78; that is what the answer really is.2164
But if we need to be able to work with something that we can actually know what the number is pretty close to,2172
because we are more concerned with having a sense of where it is than knowing precisely the right answer, then we could also get 2.79.2176
Most teachers would probably accept both; technically, √7.78 is a better answer than 2.79,2183
because we have lost a tiny bit of accuracy when we take the square root, and then we round it.2190
But we will be able to use both; they are both very good answers.2195
All right, the next one: what would be the distance between the origin and (3,12,4)?2199
So, we talked about things in two dimensions before: what if we had to deal with three dimensions, though?2206
This problem is going to happen in three dimensions.2211
We talked briefly about three-dimensional things before, where we have three different axes.2213
They are each perpendicular: so here is our x-axis; here is our y-axis; here is our z-axis; so it goes x, y, z, like this.2221
If we were to plot this point, just to try to get a sense of what is going on, then we would count out 3 on x;2234
we would count out 12 on y; and we would count out 4 on z.2242
So, our point would be out 3; from 3 out to a distance of 12; and then, from there, up 4.2248
I am starting to accidentally encroach into my words.2258
So, our line is going to look like this.2262
Now, notice, though: we could also think about a cross-section here.2266
This is getting a little complex to see: let me see if I can get it a little bit more sensible with my hands.2269
Imagine that here is our x; here is our y; here is our z.2275
So, our number is go x forward; it is going to go y 12, and it is going to go up 4 (forward by 3).2279
We go out like this: we are going to start here, and we are going to go in all of these, all at once.2286
We go out by 3; we go over by 12; and we go up by 4.2292
We could also think of this as being a cross-section: we could take a cross-section, and we could make a triangle in here.2298
We can say, "What happens in the x,y plane?": here is x; here is y.2307
We go over 3; we go up 12; and we can talk about how we got here--we can do that portion of our trip.2316
We go and travel x and y; then we go up.2327
We travel x and y; and we could figure out, "Look, the x and y plane is perpendicular to the z portion of our axis."2330
So, that is going to be perpendicular there, as well; so we can figure out what this length here is.2337
And then, we already know what this is; it has to be 4, because our height was 4.2342
If we could figure out what this portion right here is, what this length for our cross-section, the base of our triangle, is, we would be good to go.2347
We look at the x,y plane--we look at this portion here--and we can turn it into a nice, flat object that we can see--a nice planar object.2356
So, we do the same things that we have been doing before.2365
Here, 3 is our horizontal; 12 is our vertical; so our distance is going to be the square root of 12 squared, plus 3 squared.2368
Well, let's swap that around; that was exactly correct, but just to keep doing the exact same way we have been doing it precisely before.2379
We will have 3^{2}, because we had horizontal before first; and then 12^{2}, because we are always following with vertical.2387
The other way was just as right: 3^{2} + 12^{2} is the same thing as 12^{2} + 3^{2}, after all.2393
But that way, we are just following our nice pattern from before.2398
So, we can figure out what the length of this portion right here is.2402
We have it right here: we now have that--now we just bring in this thing, and we just do another one.2408
For d of our triangle here, let's use a different color; we will go with green for the distance in our three-dimensional object.2414
In our three-dimensional object, it is going to be the square root of what it was in the x,y plane--that distance,2424
squared (what was it in the x,y plane? It was the square root of 3 squared plus 12 squared; that was its distance before;2430
but we have to square it), plus...what was the jump that it had up--what was its vertical leap in the z?--that was 4, so plus 4 squared.2438
Now, notice: when we take a square root and square it, like we have in here, d =...the square root of 3^{2} plus2447
12^{2}, squared, is just going to crack it open, and we will get 3^{2} + 12^{2}, and then plus 4^{2}.2453
We simplify this, and we get d = √(9 + 144 + 16); simplify that some more; we get √169, which equals 13; so our distance is 13.2460
What we have done is: we are able to look at how it changed on the first plane.2476
We sort of take a cut, so that we can look at how it changed in the x,y plane; and then, we put on the z.2481
Now, you might be getting a sense of "Oh, maybe there is something we could do in general."2489
And I didn't tell you this before, because it is not really going to come up much in this course.2494
But we can actually get a distance for three dimensions, as well.2497
It is going to be the distance of the square root of a whole bunch of stuff, now...of (x_{2} - x_{1})^{2},2500
the square in our horizontal, plus (y_{2} - y_{1})^{2}, the square in our vertical,2510
plus (z_{2} - z_{1})^{2}, the square in our coming out of the x,y plane.2517
That comes out, because what we do is clear out the x,y plane first.2525
It is going to have a square root around it; but then, when we put in that z--when we toss out that coming out perpendicularly--2528
we are going to square root again, because now we are doing another right triangle.2536
And so, it will simplify to just each one of these differences, squared.2539
This might be a slightly complex idea for you, so don't worry if this didn't make sense.2542
Just take it out of your head; throw it away; it is not really going to come up.2546
It is just a really cool thing that...if you are thinking, "Oh, there is something interesting going on here"...you are right!2549
This is the interesting thing that is going on; we can actually generalize this to three dimensions.2553
And if we wanted, we could even keep going to 4, 5, 6...any number of dimensions we want.2558
And you might have some idea of what is going to happen as we go on to four dimensions.2562
See if you can figure out what goes on in four dimensions--it is kind of cool.2565
All right, Example 4: What is the slope between (-1,8) and (1,14)?2568
So remember, we figured out that slope is rise over run; the amount that we rise is our change in our vertical,2573
(y_{2} - y_{1})--our two vertical locations--the change--and our run is our two horizontal locations--their change.2583
So, in this case, arbitrarily, let's make this one the second one, and this one the first one.2589
There is no particular reason; it is just because that one came first, and that one came second.2595
So, let's look at it that way: it also makes sense, because if we were to draw a picture of it, we would have something like this:2600
(-1,8), and then (1,14); so it makes sense that they give this one as the second one and this one as the first one.2606
But as we will see in a little bit, it actually doesn't matter which one we choose first.2612
So, we want to find out the slope between (-1,8) and (1,14).2616
So then, m = (y_{2} - y_{1})/(x_{2} - x_{1}).2621
Our second y is 14; so 14 minus our first y (is 8), divided by our second x (is 1); our first x (is -1), so minus -1; that equals 6...2632
1 minus -1...those negatives cancel; we get 6/2 = 3, so our slope is 3.2648
What is the slope going to be if we switch our first and second points?2655
Instead, we make this one 1, and we make this one 2; well, we do the same thing: m =...2659
it is going to be...our new second one is 8, minus our new first one (is 14), divided by x_{2},2667
our new second one (is -1), minus our new first one (is 1).2681
8 - 14 is -6: -1 minus 1 is -2; will you look at that--these cancel, and we get the exact same thing.2688
If we switch our first and second points, which we arbitrarily decided to make second and first, does it affect what the slope comes out to be? No.2699
It doesn't--why? Because of the negatives: it introduces negatives on both the top and the bottom; they cancel out.2708
So, if we have negatives showing up because of the switch, they are going to show up on both the top and the bottom, so we will always see cancellation.2713
So, it doesn't matter if we plug in our one as the first one, or if we plug in that one as the second one.2719
It doesn't matter which one gets to be called first and second, as long as we match up our seconds and our firsts.2725
They have to match up vertically: if we have one point be the second point on the top, the second y-coordinate,2731
then it has to also be the second x-coordinate; it has to come first on the bottom, as well.2737
So, we have to make sure that the points match up vertically.2741
8 and -1; (-1,8); 14 and 1; (1,14); they match up there.2745
Let's prove this, though: if we want to prove that this always comes out to be the case,2753
to prove this, what we want to show is that it doesn't matter if it is (y_{2} - y_{1})/(x_{2} - x_{1}),2759
versus (y_{1} - y_{2})/(x_{1} - x_{2}); if we swap the location of which gets to come first,2771
which gets to be more on the left in the fraction, it doesn't matter which gets to be more on the left and which is more on the right.2781
That is what we want to show; so how do we prove it?2787
Well, let's start with this one here: we will have (y_{2} - y_{1})/(x_{2} - x_{1}).2789
Now, we want to be able to get that to start looking like this thing.2800
And we say, "Well, (y_{2} - y_{1})...that is pretty much the same thing, but it has a negative introduced to it."2804
So, how could we introduce some negatives here?2811
Well, let's write it again: (y_{2} - y_{1})/(x_{2} - x_{1}).2814
We could multiply it by 1, right...wait, wait, what?--yes, 1, right?--I can multiply anything by 1, any time I want.2818
You can't stop me from multiplying by 1; I can take any number and multiply it by 1, and it has no effect.2826
So, everything is equal to just itself times 1.2832
Now, the cool thing about math is that there are a lot of ways to say the number 1.2835
I can say 1 as 1, but I can also say it as 1/1; or I could say it as 5/5, or I could say it as -1/-1.2840
And that is how we introduce our negatives; and this is also the idea that is coming along when we change denominators.2849
We introduce by multiplying the same thing on the top and the bottom; we multiply by -1 on the top and -1 on the bottom.2855
Now, notice: since we are multiplying the top, we are not just multiplying the first part of the top.2861
We are multiplying the whole top; because it is multiplication, it is going to apply to this fraction as if it started in parentheses.2865
So, times -1, over -1; (y_{2} - y_{1}) times -1 becomes -y_{2} + y_{1},2872
over -x_{2} + x_{1}; and this thing right here is just the exact same thing as this thing right here.2881
We have just swapped the location; instead of -y_{2} + y_{1},2891
it becomes y_{1} - y_{2}, what we are a little more used to seeing.2895
So, we have managed to prove that it doesn't matter what order we put it into, using this (y_{2} - y_{1})/(x_{2} - x_{1}) formula.2899
It doesn't matter, because it is going to end up giving out the same answers.2905
But the really key idea to think about, when we are talking about slope, is that it is the rise over the run.2908
It is the rate of change--how quickly the line is changing.2912
All right, I hope you learned a bunch here; I hope it has been a great refresher, and everything is really understandable,2915
because we will be using these things a whole bunch, later on.2919
All right, see you at Educator.com later--goodbye!2922
1 answer
Last reply by: Professor Selhorst-Jones
Tue Nov 4, 2014 11:29 AM
Post by Jamal Tischler on November 4, 2014
How can the Pythagorean theorem pe proved ? I saw an explaination with some triangles and squares bonded, but I didn't realy understand it.
1 answer
Last reply by: Professor Selhorst-Jones
Sun Sep 21, 2014 9:46 PM
Post by Magesh Prasanna on September 20, 2014
Hello sir! asusual superb lecture...By Definition Slope =rise/run. i.e no.of rises per run. Are we only concerned about rises per run why we aren't for runs per rise?..I'm unable to imagine how line runs per rise. The rise/run of a straight line is proportinal to the angle of the line. Let me know how the value of rise/run is related to the angle of the line?
1 answer
Last reply by: Professor Selhorst-Jones
Sat Jul 5, 2014 3:56 PM
Post by Thuy Nguyen on July 4, 2014
In computer science, when implementing a binary search, using the shorter formula for finding a midpoint is wrong because it could cause an overflow of integers. I like the concise formula for midpoint, but a good reason for using the longer version: a + (b-a)/2, would be in programming a stable algorithm.
2 answers
Last reply by: Linda Volti
Fri Feb 21, 2014 6:00 PM
Post by Linda Volti on February 21, 2014
Totally agree with the first three posts: absolutely fantastic! Even though I knew most of these things, I'm now learning them at a completely different level. I wish I had a teacher like you when I was at school many years ago now!
1 answer
Last reply by: Ian Henderson
Mon Aug 12, 2013 10:45 PM
Post by Ian Henderson on August 12, 2013
Sorry I may be a bit confused here, but when we're looking for M the slope, would that not be the equivelant of looking a2+b2 = c2? The pythagorean theorem? Is C not usually the slope?
1 answer
Last reply by: Professor Selhorst-Jones
Thu Jul 11, 2013 12:16 PM
Post by Jonathan Traynor on June 26, 2013
What a perfect way to recap old material. IO love the way you appeal to intuition and then explain it in maths terms. Outstanding!!!
2 answers
Last reply by: thelma clarke
Mon Mar 16, 2015 7:16 AM
Post by Montgomery Childs on June 25, 2013
Great refresher. Love the way you break it down - 4th. dimension = t(time)?
1 answer
Last reply by: Professor Selhorst-Jones
Thu Jun 13, 2013 8:17 PM
Post by Sarawut Chaiyadech on June 13, 2013
thank you very much you make maths visible :) cheers ]