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Robert McKeown: Hello and welcome to ALEKS

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walkthrough video number eight. My name is

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Robert J McKeown. And I'm very happy to have you

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here watching my video today. our very last

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topic is topic eight geometry. Geometry plays a

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role in economics. triangles well appear a

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number of times in your undergraduate classes.

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circles are not as often, but they can be very

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useful for understanding a few important

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concepts. And economics. For example, when we

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talked about functions, the equation of a circle

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is not a function. So if you're wondering what's

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a function was not a function, the circle

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equation you're about to see is a good example

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of that. Now, some of the shapes in the ALEKS

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precalculus module and topic aid. You know, I've

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never seen a parallelogram in my economics

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career. But it doesn't hurt to go through the

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exercise of learning an equation, and then

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manipulating that equation to solve for an

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unknown variable, which is something that as an

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economist, you're going to do all the time,

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maybe we'll define a way that stock investor

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chooses their stocks. And then we use that

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expression again, and again to solve for maybe

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stock prices or something like that. So the

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exercise is still very useful. Some of the

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shapes do show up econometrics statistics, you

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can study that in a geometric approach just

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using many of the equations, or some of the

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equations anyway, that you're going to see in

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this video. So without taking up any more of

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your time, have a pencil, paper, maybe even one

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of these beauties, and wheat. Let's go ahead,

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and let's start solving some problems together.

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Let's start off by talking about circles.

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Circles come up occasionally in economics. But

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this is particularly good practice for you.

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Because you're going to be given an equation,

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and then you're asked to manipulate that

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equation. And that's something that economists

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do all the time, both in theory. And when we're

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trying to make a statistical analysis. The form

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here is very important. Our x's and y's, these

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are actually x axis and y axis coordinates. So

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those are often unknown variables. They're

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placeholders whatever the coordinates of the

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circle is, or whatever the coordinates are.

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That's where we're going to find them. What else

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is very important here are these h and k

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letters. These h and k letters give us the

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coordinates of the center of the circle. And the

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center of the circle is what we're going to need

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to graph on ALEKS. So we always need to find the

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center of the circle to graph a circle on ALEKS.

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And probably, you know, in general, if we want

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to graph a circle, knowing where its center is,

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is very helpful. Now let's take a look at our

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first question, using this equation of a circle.

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So this is the information we've been given.

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It's a good idea to answer these questions to

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memorize or know what the equation of a circle

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is. So I'm going to rewrite the equation of a

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circle up here. And notice that it's quite a bit

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different than what we've been given. So the

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question implies that we should be able to graph

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a circle here. But the way that it's given us

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the information is not in the form that we want

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it, we want to take this thing here and put it

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into that form. So how are we going to do that?

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Well, the first thing we can do is take care of

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this negative 11. And we can add 11 to both

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sides of the expression and then I'm going to

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collect, well, not so much collect like terms

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has moved like terms, or at least terms that are

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similar to each other closer together. So we're

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gonna have our X's together, x squared plus two

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x plus y squared plus, or excuse me, that's

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minus minus four y. And that's going to be equal

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to 11. Because we've added 11 to both sides of

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the equation.

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Now, we still have work to do. Because we want

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this thing here to look like that. And we want

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this thing over here. To look like that. How are

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we going to do that? Well, we're going to use a

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technique called completing the square.

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Notice that this thing up here is a square. And

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so is this over here. It's a square. That's why

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they're squared squares. And so we're going to

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complete the square and turn this expression

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into something that looks like the equation of

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the circle that we want. How are we going to do

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this? Well take a look x squared plus two x plus

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something. If I write two x plus, or I should

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say x squared plus two x plus one, that's gonna

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look like x minus minus one, squared. And now

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it's a square, we've completed the square How

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did I do that? I had to add one. Now let's take

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a look at the Y side of the expression. We've

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got y squared minus four y. And I like to

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complete the square and turn this into a squared

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expression. I'm going to add four. And if I add

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four, I can rewrite this thing as y minus two

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squared. And so what have we done, we've added a

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one, we've added a plus four, and now we've got

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11 plus one, plus four. And that's going to be

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equal to 16. And we've actually done it, I'll

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rewrite it. Notice how careful I am being about

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my signage on the left hand side.

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I'll just write 16. Over there. What do I know

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about this circle? Well, I know that age is

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equal to negative one. I know that k is equal to

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two. And so its center is negative one, two,

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there is this. Those are the coordinates of the

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center. And let me rewrite center in case you

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can read that.

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What's the radius? Well, it's equal to the

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square root of 16, which is four. So I'm capable

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now of graphing this circle. I know what its

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center coordinate is, and I know what its radius

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is. And those are the two things that ALEKS is

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going to ask us for. So let's go over to ALEKS.

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I'm going to start by clicking on the very

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circular circular, not the oval the circle. And

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we want to coordinate this at x negative one, y

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two. So I'm going to click the center of the

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circle there. And then we know the radius is

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four. And so I'm going to go 1234 spaces out, I

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could go to the left, I could go up doesn't

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matter. And I click there. And then now we've

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got our very nice circle on ALEKS. Let's take a

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look at our next circle question. We've been

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given the center of the circle. And we've been

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given one point along the perimeter of the

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circle. And the question is asking us to find

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the equation. So the equation of a circle is x

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minus h squared plus y minus k squared is equal

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to the radius squared. So we have our center so

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we have h here, and we have k here. And this

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three is our x, and this negative two is our y.

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So let's go ahead and plug those into the

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equation. When we do that. We get negative

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three. Oh, excuse me. We get three minus minus

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three squared plus

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negative two minus one is equal to r squared.

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Now, let's simplify. So we're gonna have three

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minus negative three, that's going to give us

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six squared, and we've got negative two minus

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three, which is going to be, well negative three

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squared, that's going to be equal to the radius

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squared. Working out the exponents, we've got r

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squared is equal to 45. Now that r squared is

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the only part of this equation that we're

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missing, so if we want to write out the equation

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of the circle, it's going to be x. Well, I'll

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write it as plus three squared plus y. Negative

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one square is equal to 45. I didn't even have to

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take the square root to find the radius, because

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actually, the questions not really asking us

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that. Now let's go take a look at ALEKS and see

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if we've got the right answer. I went ahead and

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input our answer into ALEKS. So you can see it

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right there. Now, I'm going to click the check

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button. And let's see if we've got the right

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answer. And we do. We're asked to calculate the

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distance between the point H and the point F.

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And the trick here is that H and F don't have

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the same y axis value, and they don't have the

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same x axis value. So we've got a diagonal line

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here, it's not horizontal, it's not vertical, it

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was horizontal, the distance would be easy. Was

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vertical, the distance would be easy. So how do

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we measure the distance when we're moving across

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the x and y axis? And the answer to that is

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going to be, we're going to use a distance

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formula. So let's take a look at the nodes.

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You're likely already familiar with this

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expression. And it is something like this. The X

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distance plus the distance along y, both of

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which are squared, and then we'll take the

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square root of the sum of both notice

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these two are not equal. So we can't do that

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that is wrong. Don't do that, we're going to

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have to deal with the squared terms first. And

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then once we take care of the squared terms, sum

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everything up, and then we can take the square

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root. So looking back at the question, let's

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call this observation x two, and that eight will

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be our y two. So H is going to be our second

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observation. And we'll let f be our first

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observation. So we'll call this x one. We'll

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call that x two. Now, all we have to do is plug

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in the numbers. So we've got minus nine minus

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minus two. And that whole thing squared plus

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eight minus four, that whole thing squared. And

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then not to forget, we're going to take the

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square root of the whole thing, and that's going

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to be equal to D. Now let's work through some of

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these numbers. So we've got negative seven

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squared plus four squared is equal to something

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that's going to be 49 plus 16, which is, Oh,

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don't forget my square roots.

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And we've got the square root of three 65 Now

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let's see if we've got the right answer on

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ALEKS. Here we are on ALEKS. I'm going to click

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this little symbol over here the square root

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sign in the 65. Notice that the question wants

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us to give an exact answer. We're not allowed to

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give a decimal approximation. So I'll leave it

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with the square root above it. And I'll click on

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the check button and we got the right answer. So

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the distance between point H and point F is the

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square root of 65.