WEBVTT mathematics/geometry/pyo
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Hello; welcome to Educator.com.
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This is the Geometry course; the very first lesson is on the coordinate plane, which should be somewhat of a review.
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So, make sure to check out the other free lessons of the syllabus.
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Let's begin: the **coordinate system**: the coordinate plane is part of the coordinate system.
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This here is called the **coordinate plane**; right here, this is the **x-axis**, and this is the **y-axis**.
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And these two make up the four **quadrants** of the coordinate plane.
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If we were to label these, this is I, II, III, and so on; we know that if we go this way, we are going to be going negative; positive and negative.
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Now, these axes make up four quadrants; the four sections of the coordinate plane are known as **quadrants**.
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Here, this is the first quadrant, so this is quadrant I; around this side, we have quadrant II; this is quadrant III, and quadrant IV.
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So, it starts here, and it goes this way: I, II, III, IV.
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And for quadrant I, we have a positive; we are only dealing with the positive x-axis and the positive y-axis.
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For quadrant II, we have a negative x-axis, and then the positive y-axis.
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For quadrant III, it is negative x and negative y; quadrant IV is positive x and negative y.
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Those are quadrants; make sure that you remember that there are four of them.
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The **origin** is right there: this is known as the origin.
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The origin is (0,0): the x is 0, and the y is 0--right where they meet, that is the origin.
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And this is an example of an **ordered pair**: an ordered pair is when you have the x-coordinate paired with a y-coordinate.
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And together, it is called an ordered pair.
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So, if I have a point, (4,2), this would be an ordered pair; my x-coordinate is before, and then 2 would be my y-coordinate.
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So, let's practice graphing using the coordinate plane: we are going to look for these points and write the coordinates.
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Here is A, B, and C; for point A, we know that this is 0; this is x; this is y; here is 1, 2, 3, -1, -2, -3;
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so for point A, we always start with the x-axis first; so the x-coordinate goes first, and that is 1;
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and then what is my y? 1; that is my ordered pair.
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For B: my x-coordinate for point B is -1, and my y is going to be -2; so here is (-1,-2).
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And for C, it is 2 for my x and -1 for my y.
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We are going to graph each of the points on the coordinate plane.
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For A, I have (4,2); this is my x and this is my y; this is my x and this is my y.
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So, I am going to go to positive 4 on this side: 1, 2, 3, 4; and 2 on my y is +2, which is there; (4,2) is going to be right there.
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I am going to label that point A.
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For B, (-3,0): x is -3, which is 1, 2, -3...and my y is 0; that means I do not go up or down anything--I stay right there.
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OK, this is where y is 0; and this part right here is going to be B.
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C: it will be 1, 2, and 1 right there for C; and then D is (-4,-2); OK.
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OK, there are all my points on the coordinate plane.
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One more thing to go over: **collinear** points are points that lie on the same line.
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When we have points that line up--you can draw a line through those points--those points will be collinear.
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And let's see if we have any collinear points here.
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Well, if you remember from algebra, for slope, we have rise over run;
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all of those have to do with a line and points, when we are graphing lines on the coordinate plane.
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So, here we have A and C--we know that those two, or any two, points will be collinear,
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because you can draw a line through any two points.
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Here, I know that those three points, A, C, and D, are collinear, because they will be on the same line.
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They lie on the same line, so A, C, and D are collinear.
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If you want to double-check this, you can use what you learned from algebra; you can count rise over run.
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Find your slope from D to C, and then from C to A; and it should be the same, and also from C to A and D to A.
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OK, so points A, C, and D are collinear.
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Write the coordinates and quadrants for each point: let's look at point A.
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Point A: this is -1, and my y is -1, -2, and -3; so for A, -1 is my x-coordinate, and -3 is my y-coordinate.
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And this is quadrant III, because it goes I, II, and III; so this is in quadrant III.
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For B: my x is +1, and my y is +2; and that is in quadrant I.
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C is 3, and my y is -1; and that is quadrant IV; D is -3, and 3; quadrant II.
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OK, let's do another example: Name two points in each of the four quadrants.
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OK, we have quadrant I; now quadrant I, I know, is here; quadrant II, quadrant III, and quadrant IV.
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Quadrant I is going to be positive, and then my y-coordinate is going to be positive.
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Quadrant II: my x (x is always first), we know, is negative; and then y is positive.
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Quadrant III is negative for the x-coordinate and negative for my y.
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Quadrant IV is positive and negative.
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They should all be different; their signs will be different for each of the quadrants.
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So, I can name any point; as long as my x-coordinates and my y-coordinates are both positive, they are going to be from Quadrant I.
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I can just say (1,2) and then maybe (3,4); those are two points from Quadrant I.
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Quadrant II will be...we have to have a negative x-coordinate and a positive y-coordinate, so what about (-1,2) and (-3,4).
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Now, you can use your own numbers; you can use the same numbers.
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As long as you have a negative x and a positive y, they are from Quadrant II.
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Quadrant III: x and y are both negative, so (-1,-2) and (-3,-4) will be from Quadrant III.
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And then, Quadrant IV: we have a positive x and a negative y, so (1,2) and (3,-4).
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Those are two points from each of the four quadrants.
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The next example: Graph each point on the same coordinate plane.
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Let me do these: the first one, point A, is (0,3).
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Now, be careful--this 0 is my x; that means, on my x-axis, I am going to be at 0, which is right there.
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And then, for my y (I'll just write out a few of these numbers: 1, 2, 3, 4, 1, 2, 3...OK, let me erase that...-3 and -4; OK)...
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again, it is 0 for my x, and then 3 on my y; so there is 3 on my y.
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And that is going to be my point A.
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For point B, I'll go to -2 on my x and -1 on my y; there is B.
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C is -5; there is -5 on my x and 0 on my y; that means I am not going to move up or down; I am going to stay there; there is C.
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And then, D will be 4, and then -6 is all the way down here; so there is point D.
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And my final example: Point A is (3,1) and B is (0,-5); they both lie on the graph y = 2x - 5.
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Determine whether each point is collinear with points A and B.
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OK, if I have my coordinate plane, my x- and my y-axis, my point A is going to be (3,1); there is A.
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B is going to be (0,-5), right there; there is B.
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They both lie on the graph y = 2x - 5; so if I draw a line through these points, that is going to be the line for this equation of y = 2x - 5.
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And you are just going to determine whether each point is collinear with the points A and B.
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Now, "collinear" means that they are going to be on the same line.
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So, we are just going to see if these three points (since we know that points A and B are on this line) are going to also be on the line.
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And if they are, then they will be collinear with the points.
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For point C, instead of graphing the line and seeing if the point lies on the line, you can just...
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since you know that this is x and this is your y, you can just plug it into the equation and see if it works.
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y = 2x - 5: you are just going to plug in -1 for x and 4 for y.
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So, 4 = 2(-1) - 5: here, this is 4 = -2 - 5; do we know that...since we don't know that these are equal...does 4 equal -7?
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No, it does not; so this point does not lie on this line; that means that point C is not collinear--this says no.
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OK, for point D, I am going to also plug in: 7 is my y; 7 = 2(6) - 5.
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OK, I am going to put a question mark over my equals sign, just because I am not sure if it does yet--I can't see if it equals.
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This is 12 - 5; 7 = 7, so this is a yes--they are collinear points.
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And then, my last point, point E: -15 = 2(-5) - 5): put a question mark again.
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-10 - 5...-15 does equal -15, so this is also a yes; OK.
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Points D and E are collinear with points A and B, since they are all on the line y = 2x - 5.
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That is it for this lesson; thank you for watching Educator.com!