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Midpoints, Distance, the Pythagorean Theorem, & Slope

• To find the midpoint in one dimension, we take the average of the two numbers involved:
 a+b 2 .
• To find the midpoint in two dimensions (in the plane), we take the average location for each dimension on its own:
 ⎛⎝ x1 + x22 , y1 + y22 ⎞⎠ .
• 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
 a2 + b2 = c2.
• 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 = √ (x2 − x1)2 + (y2 − y1)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 = riserun = vertical changehorizontal change = y2 − y1x2 − x1 .
• 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

What is the midpoint between 5 and 47?
• The midpoint formula (in one dimension) is
 a+b 2 .
• Set one of the numbers as a and the other as b, then use the formula.
26
What is the midpoint between 8.5 and −15.3?
• The midpoint formula (in one dimension) is
 a+b 2 .
• Set one of the numbers as a and the other as b, then use the formula.
−3.4
What is the midpoint between (5, −18) and (7,  14)?
• The midpoint formula (in two dimensions) is
 ⎛⎝ x1 + x22 , y1 + y22 ⎞⎠ .
• The x-coordinates of the ordered pairs give x1 and x2. The y-coordinates of the ordered pairs give y1 and y2.
• Plug in to the formula and simplify.
(6,  −2)
What is the midpoint between (−6t,  10) and (−4t,  −3)?
• The midpoint formula (in two dimensions) is
 ⎛⎝ x1 + x22 , y1 + y22 ⎞⎠ .
• The x-coordinates of the ordered pairs give x1 and x2. While we have variables in the x-coordinates for this problem, it has no affect on setting them as x1 and x2.
• The y-coordinates of the ordered pairs give y1 and y2.
• Plug in to the formula and simplify.
(−5t,  3.5)
What is the distance between the numbers 3 and 50?
• The distance formula (in one dimension) uses absolute value:
 |a−b|.
• Absolute value "forces" whatever is inside to be positive. Negative numbers are turned positive, while positives are not affected (since they're already positive).
• Set one of the numbers as a and the other as b, then use the formula.
47
What is the distance between the numbers 84 and −46?
• The distance formula (in one dimension) uses absolute value:
 |a−b|.
• Absolute value "forces" whatever is inside to be positive. Negative numbers are turned positive, while positives are not affected (since they're already positive).
• Set one of the numbers as a and the other as b, then use the formula.
130
What is the distance between (8,  −3) and (5, 1)?
• The distance formula (in two dimensions) is
 √ (x2 − x1)2 + (y2 − y1)2 .
[Note: If you forget this formula, you can set it up by creating a right triangle with the hypotenuse going between the two points, then using the Pythagorean theorem.]
• The x-coordinates of the ordered pairs give x1 and x2. The y-coordinates of the ordered pairs give y1 and y2.
• Plug in to the formula and simplify:
 √ (5 − 8)2 + (1 − (−3))2
5
What is the distance between (7,  11) and (20, 9)?
• The distance formula (in two dimensions) is
 √ (x2 − x1)2 + (y2 − y1)2 .
[Note: If you forget this formula, you can set it up by creating a right triangle with the hypotenuse going between the two points, then using the Pythagorean theorem.]
• The x-coordinates of the ordered pairs give x1 and x2. The y-coordinates of the ordered pairs give y1 and y2.
• Plug in to the formula and simplify (you'll probably want a calculator):
 √ (20 − 7)2 + (9 − 11)2
√{173} ≈ 13.153
What is the slope of a line that goes through (1,  7) and (4,  1)?
• Slope is symbolized by the letter m, and defined as
 m = riserun = vertical changehorizontal change .
This gives us the formula
 m = y2 − y1x2 − x1 .
• The x-coordinates of the ordered pairs give x1 and x2. The y-coordinates of the ordered pairs give y1 and y2.
• Plug in to the formula and simplify.
m=−2
A line goes through the points (5,  5) and (1,  −1). If the line also goes through the point (7,  a), what is the value of a?
• Use the first two points to find the slope of the line:
 m = y2 − y1x2 − x1 .
[See the previous question if you have difficulty with this.]
• We find that the line has a slope of m=1.5.
• Using this slope and one of the other points on the line, we can find what the value of a must be.
• One way to do this is by thinking in terms of "steps". We start at (5,  5). Notice that (7,  a) is two horizontal "steps" away. Thus, we have two vertical steps of size m = 1.5, which will give a total rise of 3. Alternatively, we can just choose one of the previous points on the line and set up the slope formula to solve for a:
 m = y2 − y1x2 − x1 ⇒     1.5 = a − 5 7 − 5
a=8

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

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 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

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, (x1,y1) and (x2,y2)?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: y2 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 (x2,y1).0498

So, the midpoint horizontally is going to be (x2 + x1)/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 (y2 + y1)/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 x1 + x2, so its midpoint is (x1 + x2)/2.0542

And vertically, our locations were y1 and y2, so the middle of our vertical locations will be (y1 + y2)/2; great.0548

Our midpoint formula is just (x1 + x2)/2, and (y1 + y2)/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, a2 + b2...each of the smaller legs squared,0849

then added together...that is going to be equal to our hypotenuse, squared.0854

a2 + b2 = c2: 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 d2, the diagonal, the hypotenuse,0957

is going to be equal to 62 + 82 = 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--(x1,y1), our first point, and (x2,y2).0985

Now, I didn't talk about this explicitly the first time, but when I say x1, 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--(x1,y1),(x2,y2)--0996

that is what you should interpret when you see those little subscripts, those little numbers on the bottom right.1003

So, (x1,y1), (x2,y2): 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 x2 (because it is going to have the same horizontal location as our second point),1040

we went from x1 to x2; our distance is the absolute value of (x2 - x1).1047

The horizontal length is going to be the absolute value of (x2 - x1).1053

What is the vertical length--what is the vertical leg of our triangle?1056

Well, we end up at y2; and what is the location that we are starting on this triangle?1059

It is going to be y1, because it is going to be the same as over here.1064

So, we take y2 - y1; 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 (y2 - y1).1074

So, if we want to know what the distance of the diagonal is, it is going to be d2 = (|x2 - x1|)2 + (|y2 - y1|)2.1078

Now, there is a little thing that we can notice at this point.1096

OK, if I have |x2 - x1|, and then I square it, well, if I just take1099

x2 - x1, 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 x2 - x1, squared, is equal to the quantity1130

(x2 - x1), 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 d2 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 ((x2 - x1)2 + (y2 - y1)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, (x1,y1) and (x2,y2).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 (x2 - x1).1388

That is how much we changed as we went from left to right.1395

And the amount vertically that we changed is (y2 - y1), because we went up from y1 to y2).1398

We went up from y1 to some y2; so it is y2 - y1.1405

So, our rise is y2 - y1, and our run is x2 - x1.1411

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 (y2 - y1)/(x2 - x1).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 (y2 - y1)/(x2 - x1).1440

because (y2 - y1 is how much we rose, and (x2 - x1) 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 = √[(x2 - x1)2 + (y2 - y1)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 (x2 - x1)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 (x2 - x1)2 is the same as (x1 - x2)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 32, because we had horizontal before first; and then 122, because we are always following with vertical.2387

The other way was just as right: 32 + 122 is the same thing as 122 + 32, 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 32 plus2447

122, squared, is just going to crack it open, and we will get 32 + 122, and then plus 42.2453

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 (x2 - x1)2,2500

the square in our horizontal, plus (y2 - y1)2, the square in our vertical,2510

plus (z2 - z1)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

(y2 - y1)--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 = (y2 - y1)/(x2 - x1).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 x2,2667

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 (y2 - y1)/(x2 - x1),2759

versus (y1 - y2)/(x1 - x2); 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 (y2 - y1)/(x2 - x1).2789

Now, we want to be able to get that to start looking like this thing.2800

And we say, "Well, (y2 - y1)...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: (y2 - y1)/(x2 - x1).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; (y2 - y1) times -1 becomes -y2 + y1,2872

over -x2 + x1; 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 -y2 + y1,2891

it becomes y1 - y2, 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 (y2 - y1)/(x2 - x1) 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