Common Core Geometry Unit 5 Lesson 1 Slope and Parallelism

Determine the slope of both of these lines graphically, recall that slope is the ratio of the rise of the line to the run of the line as we move from right to left along the X axis. Now, this is exceptionally important. Whenever you read a graph, read a graph. You want to read it going from left to right. And what that will always ensure is that the run, the horizontal change in the line, is positive. Therefore, we only have to worry about whether the rise is positive or negative. Let's jump right in and take a look at segment AB. Now we see automatically that AB is rising as we go from the left to the right, and therefore should end up having a positive slope. But let's calculate those rise and run. All right? Let's get ourselves a little bit of color in here. Okay. And really, what we're thinking about is as we move from point a to point B how much horizontal distance do we move, that's the run, and how much vertical distance do we move? That's the rise. In this case, we can simply count and the rise is 5. And the run is ten. So if we come over to here, we can simply say, ah, our line went up 5 units between a and B, it ran ten units, and the ratio of the rise to the run is 5 to ten. Which we can reduce by dividing both numerator and denominator by 5 to get one half. 

Now, one of the neat things about reducing slopes down besides the fact that you can see them in their simplest form is that you can also see that reduced form literally on the graph. Let's take a look at the graph one more time and see if we can zoom in on it. Hopefully we can. Sometimes that's not so easy with the board. As it's not turning out to be here. All right. We're not going to zoom in apparently. But let's take a look at something. All right, remember, we reduced that slope down to one half, and what we can see is that if we just take a look at all the points that lie at integer locations along this line, to go from one point to another, we're moving two to the right and one up. We can see that reduced slope in that two movement to the right, one movement up. The ratio of rise to run. Let's take a look at the other line segment. Now immediately we see that line segment CD is falling as we move from point C to point D again, always read that graph from left to right. But let's get those rise and run numbers. Okay. Now, in order to go from C to D, I think we'll just stick in red here to contrast the two lines. We need to run a total of 8 units, but we don't have a rise now. 

We technically have a fall. And what do we have? We've got three, 5, 8, and 12. I'm going to actually label it as negative 12 right there. Somehow I've lost my pen. Let's get it. There we go. Negative 12. So our rise is a negative 12 units are run is a positive 8 units, so the ratio negative 12 to 8 is the slope. We can again reduce that down. That ends up being a slope of negative three halves. In simplest form. Again, what that means is that for every two units we move to the right, we will drop three units in the vertical direction. Slope is always telling us a direction that we move. In terms of how much right left and how much up down. Let's go on to the next topic. We don't want to always have to calculate slope graphically. It's nice when we can. But we want to be able to do it if we simply know the coordinates of the points that we're looking at. So if we know that we have two points generically labeled as X one Y one and X two Y two, we can calculate the slope of this famous formula. The slope is the change in Y over the change in X, Y two minus Y one, divided by X two minus X one, or you can flip flop those points, just make sure to keep the rise in the numerator and the run in the denominator. 

Make sure you understand why the slope formula works. Never let a formula just kind of sit there without understanding it. You see what we really have going on here is we have a right triangle. The run is given by simply taking these two X coordinates and subtracting them. And the rise is given by taking the two Y coordinates and subtracting them. This allows us to calculate the slope of a line segment if we know it's two endpoints without having to plot the line segment. And let's do that a little bit in the next exercise. All right, real kind of plug and chug exercise here. We've got three different situations where we've got coordinate points and we want to calculate the slope. Let me read the problem to you. Exercise two, using the slope formula only. Find the slope of the line segment that has the following endpoints. Write your slope in simplest form. All right. Let's jump in. I'm going to do letter a, then I'm going to have you work on letters B and C on your own. And then we'll go through them. First, let me go back to blue ink. And let's do this. Simple enough, right? It never hurts to write down a formula when you're first using it. Slope, which we for some almost inexplicable reason, give the letter M I think of that as M for movement. Is Y two minus Y one over X two minus X one. 

It doesn't matter which of the two points you designate as X one, Y one, and X two, Y two, but it'll be natural to think about that first one as X one, Y one, and the second one is X sorry, as X two, Y two. Let's get back into the calculation. So here we go. If this is X one, Y one, and this is X two, that almost looks like a two, but not really. Y two, then we can write down our calculation as follows. Ten minus four, all divided by 8 minus, oh, be careful here, negative two. Now ten minus four simple enough is 6. Don't forget that when you subtract a negative, it becomes a positive. So 8 minus negative two is going to be positive ten. We reduce that to simplest form and we get a slope of three fifths. Indicating again that for every 5 units we move to the right, we're going to move three units up along this line segment. Now what I'd like you to do is take a few minutes, pause the video if you need to. Calculate the slopes and letter B and C and then unpause the video when you want to see the work done by me. All right, let's go through letters B and C I'm not going to write the formula out again. I'm just going to do the calculation in letter B, we have the slope is Y two, negative 9. Minus Y one, three, divided by 11 minus negative ten. 

All sorts of things can trick you up on things like this with the sign numbers, negative 9 minus three is negative 12. And 11 minus negative ten is positive 21. If we reduce that by dividing both numerator and denominator by three, we'll get a slope of negative four sevenths. Let's do letter C in this case, our slope will be Y two minus Y one, three -11. Divided by negative two minus two, be careful. That's not going to be a zero in the denominator. Not yet at least. Three -11 is negative 8. Negative two minus two is negative four. Negative 8 divided by negative four is positive two. A lot of people like to express their slopes as fractions so that they can think about rise and run. Obviously two is just an integer and you can lead the slope is two, or you can express it as two divided by one. Thus being able to think about it is moving one unit to the right while moving two units up along this line. It is critical in this unit that you're able to accurately and quickly calculate slopes, not making common mistakes like putting the change in X in the numerator and the change in Y in the denominator. And also all of the little numerical mistakes that can happen was signed numbers. B good about these calculations. If you have to use your calculator, bring it in to check your work at the very least. Let's move on to the next exercise. All right. Here, we want to make the connection between slope and parallel lines. It's an easy connection. 

It's one that you already know in all likelihood, but it's one that's going to be very important in this unit. So let's take a look at it. In exercise number three, it says, in the diagram below, the two pairs of parallel lines are shown. Letter a asks us to name the parallel line pairs. I bet you can do this for yourself. So take a moment and try to name the two pairs of lines that are parallel to one another. All right, let's go through it and make sure that we review how to use the parallel line symbol. It's pretty obvious to me that line M is parallel to line S so that's one pair. And line N is parallel to line R that's the other pair, right? But now we want to make the connection to slope. And that's going to occur in letter B in letter B, it says for each pair determine the slopes that the two lines that make up the pair. You can do this graphically or algebraically. For me, because I have the graph simply sitting here. I'm going to do it graphically. And I'm going to do both of them together, but take a minute and try to figure out the slopes of both of the lines for yourselves before we pick it back up. All right, let's get back into it. So let's do line M, okay? If we look at line M and we pick any given easy point on it. What we can see is that we need to go three units to the right and two units up each time on that line. 

At least in order to go between two nice points. And therefore, it's slope, rise over run. Uh oh, M and M, that's okay. We'll live with that. It's going to be two thirds. Let's take a look at the slope of line S, the line that it's parallel to. What we see with line S is that we also move three units to the right. And two units up between each of the two points. That are consecutive on the line. And so its slope is also two thirds. All right? Let's take a look at lines and in R now again, I hope that just by looking at the picture, you can tell that lines M and S have positive slopes and lines and an R have negative slopes. But let's do N and R for line N, what we find is to go from this point to this point, we would move two to the right, but we would have to go 5 down. So its slope is going to be negative 5 halves. On the other hand, oh, that was actually line R my apologies. Let me erase that. That was line R line N well, it looks like it's the same thing. We're going to go two units to the right. 5 units down. So its slope is also negative 5 halves. All right, now, of course, the most important connection of the lesson is in letter C let me read that for you. In letter C, it says from this exercise, what seems true about two lines that are parallel. Well, see if you can write down an answer. Just from this exercise, what do you think is true about two parallel lines? Well, it appears that two parallel lines have the same slope. So let's write that down. Two lines. That are parallel. 

Let's use a nice mathematical term for it, have equal. Slopes. At least that's what it appears. And that is, in fact, true. Let's summarize on the next slide. Slope and parallelism. Two non vertical lines will be parallel if and only if they have the same slope. Now a couple little explanations in that mathematical jargon. Number one, why do I have to throw this in? To non vertical lines. Well, what we'll see a few lessons from now is that you can't quantify the slope of a vertical line. So you can't really say the two vertical lines have equal slopes because vertical lines won't have a slope. But don't worry about that for right now. That's kind of a side case. We don't have to worry too much about vertical lines. But then the if and only if piece. Let's talk about that just for a moment. What that means is that parallel parallelism and slope to mighty big word parallelism. Parallelism and slope, there are two way road, meaning that if two lines are parallel, they have the same slope. And if two slopes are equal, then that indicates parallel lines. It works both ways. Again, if I've got this blue line and this red line that are parallel, and this one has a slope of one third, then this one somewhere will also have a slope of one third, or maybe it won't tear it as. I couldn't find the hidden slope. Slope will tell us whether things are parallel or not. And let's see that in the last exercise. 

All right, exercise number four. Given points a, B, C, and D, do the following. Letter a asks us, is AB parallel to CD. Give evidence to support your answer. Well, for us, it's never going to be sufficient to just draw something out on the coordinate plane and say, well, they look parallel. We're going to have to use calculations the tool of the slope calculation in order to see whether or not these two line segments are parallel. But that simple enough. What I'd like you to do is take a moment to calculate the slope of those two line segments and see what you would conclude. All right. Let's get into it. Let's make sure we're very good about our work. We lay it out. So everybody knows what we're doing. Line segment a, B let's do it. The slope of line segment AB will be 7 minus one, divided by 6 minus negative two. That's going to give us a 6th in the numerator. And 8 in the denominator for a reduced slope of three fourths. Let's take a look at CD. The slope of CD is going to be 6 minus. We have to be careful here. 6 minus negative three. Divided by 8 minus negative four. 6 minus negative three is positive 9, 8 minus negative four is positive 12. And if we reduce that, it becomes three fourths. I want to step back just for a moment and talk about this. This is one of the primary reasons we want to reduce fractions, especially when we're doing slope calculations. Initially, a slope of 6 8 and 9 12, they may not look like they're equal. But when we reduce them both to three fourths, we realize those two ratios are the same, and we can now say yes. Yes, guess what? Yes, AB is parallel to CD. Because slopes are equal. Little abbreviation that's completely okay. 

Now we're asked the same thing in letter B except for two different line segments. Letter B asks us is AC parallel to BD. Again, what I'd like you to do is pause the video right now and work through it to see what you find. Okay, let's go through the calculations. Now we have to be a little bit careful because now we're calculating the slope of AC and points a and C aren't sort of sitting right beside each other in the problem. But that's simple enough. We just have to keep track of everything. We're going to have a slope of negative three minus one. All divided by negative four minus be careful, negative two gives us negative whoops, gives us negative four in the numerator. And it gives us negative two and two in the denominator, which gives us a positive two when we reduce it. On the other hand, BD, let's take a look at it. It's going to have a slope of 6 -7 in the numerator. And 8 -6 in the denominator. Which is going to give us negative one in the numerator and two in the denominator. These two slopes are equal. And again, the connection between slope and parallelism is solid. If these two slopes aren't equal, then these two line segments can't be parallel. So absolutely not. No. Slopes. Are not equal. Okay. Now, the final part of this problem is the most challenging one, because in parts a and B, all we had to do is do a couple slope calculations and then compare the numerical results we got to see if the two line segments were parallel. But let's take a look at letter C if point E exists such that its X coordinate is 12 and ED is parallel to AC, then what is the Y coordinate of point E show how you arrived at your answer?