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Vertical and Horizontal Lines
Vertical and horizontal lines. As we discussed in the previous video, every line in the xy plane has its own unique equation. The simplest equations are those for horizontal and vertical lines. Let’s think about a typical horizontal line. So, there, we have a typical horizontal line.
Here are some of the points on the line. And of course these are just a few because there are also all the fractions in-between. But notice the basic pattern. The x coordinate could be almost anything. In fact the x coordinate could be any real number.
But the y coordinates, those are all the same. So, every y coordinate has to be negative 3. A very elegant way to state this condition is simply y equals negative three. That’s the equation of the line. So that specifies the condition, that the x can be whatever it wants to be, but the y has to equal negative three.
Similarly, any horizontal line will be composed entirely of points at the same height. That is, the same distance above or below the x axis. If we simply specify that height, the place where the horizontal line crosses the y axis, then we specify everything about it. Thus, the general form of any horizontal line is y = K where K is the height of the line.
K is, in fact, the y intercept. For example, this line, which intersects the y-axis at two must have the equation y equals two. That equation specifies everything about that line. What is the equation of the x-axis? Hm. Think about this, the x-axis is a horizontal line.
So it must have it’s own unique equation, just like every other line in the x-y plane. The x-axis is a horizontal line with a height of zero, because it passes through the y-axis at zero. So that means its equation must be y equals zero. That is the equation of the x-axis.
Now let’s look at vertical lines. Just as horizontal lines have all the same y coordinates. We looked at all these points on this vertical lines, they’d have al the same x coordinates. So the points here would be (4,0), (4,1), (4,2), (4,3) 4 negative 1, 4 negative 2, 4 negative 3 and so forth.
The y coordinates could be any real number on the number line. The x-coordinate has to equal four. Every point on this line has an x-coordinate of four. Thus it’s equation must be x equals four. The equation of any vertical like that passes through the x axis at K must be x = K.
Similarly, the equation of the y axis, that’s a vertical line that intersects the x axis at 0, must be x = 0. And in fact, x = 0 is the equation of the y-axis. Any two points that share the same y-coordinate must lie along the same horizontal line. Any two points that share the same x-coordinate must lie along the same vertical line.
If point C Has the same x-coordinate as point A, and the same y-coordinate as point B, then we’d know that angle ACB is a 90 degree angle. It’s a right angle. Keep in mind that a horizontal line can go through quadrants one and two, or it can go through quadrant three and four.
A vertical line can go through three and two, or it can go through four and one. Most horizontal and vertical lines move through two quadrants. As we will see, most slanted lines or oblique lines move through three quadrants. Here’s a practice problem. Pause the video and then we’ll talk about this.
Okay. A rectangle is formed by lines y equals one. Y equals four, that’s the bottom and top of the rectangle here. X equals two, that’s the left side of the rectangle, and line D, which is the right side. When the diagonal is constructed it makes an angle 30 degrees with the base.
Find the equation of line D. Very interesting, we have to remember some geometry here. So we know that line D is a vertical line, I’m just gonna say call it X equals K. We don’t know the coordinate right now,just call it K. So look at the triangle and give the vertices letter names. And so, of course K will be in the coordinates of B and C.
And of course, this is a 30, 60, 90 triangle. So the ratio of AC over three is going to be rate three over one, because those are the ratios of a 30, 60, 90 triangle. So if we multiply, cross multiply, we get AC equals three root three. And so we know that the length of that base, AC, is the difference in the x coordinates.
So it’s k minus two. But we just found that AC is three root three, so k minus two must equal three root three, and k must equal two plus root three over three. And that’s the value of k and so the equation of line D. Or we could also call it the equation of line BC. This is x equals two plus three root three.
In summary, horizontal lines have the general form y equals k. Vertical lines have the general form x equals k. The x axis is y equals zero. The y axis is x equals zero. If two points share the same x-coordinate, they are vertically separated. And if two points share the same y-coordinate, they are horizontally separated.
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