In the last video we looked at graphing lines. No we’ll dig a bit deeper into vertical and horizontal lines, and how we can find their equations.
Transcript: Vertical and Horizontal Lines
As we discussed in the last video, every line in the x-y 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 here’s the horizontal line and I highlighted the points on the line. Let’s just think about the coordinates of these points.
Some of the points on the line are things like (0, -3), (1, -3), (2, -3) etc. -1, -3, -2, -3, etc. The x-coordinate can be any number on the number line, it can even be fractions. I haven’t listed those, but it can be fractions as well. But notice that the y-coordinate is locked in place. The y-coordinate has to be negative 3.
Well, a very elegant way to state this condition is simply y = -3. That’s the equation of the line, that sums up everything you need to know about the line. In order to qualify as a point on that line, that point has to have a y coordinate of -3 and the x coordinate can be whatever it wants. And that’s the equation of the line.
The General Form of a Horizontal Line
Another way to think about this, 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 the horizontal line is y = K where K is the height of the line.
And K would also be the exact point on the y-axis, where the line crosses the y-axis. We call that the y intercept. We’ll be talking about that more in upcoming videos. For example this line here ever single point has a y coordinate of 2 and it crosses the y-axis at 2. So must have the equation y = 2.
What Is the Equation of the X-Axis?
What is the equation of the x-axis itself? Now that’s interesting. The x-axis is a horizontal line, so must have its own unique equation. Every line in the x y plane has it’s own unique equation while the x-axis is aligned so must have its own unique equation. Now, let’s think about this the x-axis is a horizontal line with a height of zero.
Because it passes through the y axis at the origin, it passes through at zero, zero. So that means that the equation of the x-axis must be y = 0, and that is the equation of the x axis.
Now let’s talk about vertical lines. Just as horizontal lines have all the same y coordinates, vertical lines have all the same x coordinates.
So here, we see that the line crosses the x axis at 4. We also see that all the points above and below it are at that the same distance to the right of the y axis. So they all have to have an x-coordinate of 4. And therefore a good way to write the equation of that line will just be x=4 rather the words that rule that we are setting for that line is the x-coordinate has to be 4.
The y-coordinates can be wherever it wants and if we follow that rule, we’ll always land on this particular line. The equation of any vertical line that passes through the x-axis at K must be x=K. Similarly, the equation of the y-axis, a vertical line that intersects the x-axis at zero, must be x = 0.
Two Points that Have the Same X or Y Coordinate
So the equation of the x-axis is y = 0, the equation of the y-axis is x = 0. Any two points that share the same y-coordinate must lie along the same horizontal line. That is a very important idea. And that’s one you need to recognize because the test will just give you a bunch of coordinates.
You’ll have to recognize these two coordinates have the same y-coordinate. So they must be on a horizontal line. Similarly, any two points that share the same x coordinate must lie along the same vertical line. If C has the same x coordinate as point A and the same y-coordinate as point B, then it must be true that the angle, ACB, is a right angle, a 90-degree angle.
Because it is the angle between a horizontal line and a vertical line. Keep in mind that a horizontal line can go through the first and second quadrants, that is if it’s above the x axis, or it can go through the third and fourth quadrant if it’s bellow the x axis. A vertical line can go though quadrants two and three if it’s to the left of the y axis, or it can go through quadrants four and one if it’s to the right of the y axis.
Most horizontal and vertical lines move through two quadrants. As we will see, most slanted lines move through three quadrants.
Vertical and Horizontal Lines: Practice Problem
Here’s a practice problem.
A rectangle is formed by the lines y = 1, y = 4, x = 2 and line = D. When the diagonal is constructed it makes an angle of 30 degrees at the base. Find the equation of Line D.
So I’m going to suggest that you pause the video, work on this, and then we’ll talk about it.
All right, well, first of all, it’s obvious that D is a vertical line. Like all vertical lines, it has to have an equation of the form x = K. So let’s think about this, we’re gonna look at the triangle and give the vertices letter names.
So the point at 2, 1 we’re gonna call that A, the point at K, 4 we’re gonna call B and the point at C, at K,1 we’re gonna call C. And notice that the length of that base is K- 2 because from starting out from the y-axis we’d move two spaces to the right to get to A then when we move out to C we’ve gone K spaces. So the little 2 plus the bottom leg of this triangle add up to K.
That must mean that the base is K- 2. So, of course this is a 30-60-90 triangle. We studied this in the geometry lesson and we know that we can set up a ratio, A over C = root 3 over 1. Cross multiply, we get AC = 3 root 3. And so this AC, which is 3 root 3, this equals K- 2, as we’ve said.
So now we have to add 2 to solve K, and we get K = 2 + 3 root over 3. And therefore the equation of the line is x = 2 + 3 root 3.
In summary, horizontal lines have the general form y = K. Vertical lines have the general form x = K. The x axis has the equation y = 0.
The y axis has the equation x = 0. If two points share the same x-coordinate, they are vertically separated. They lie on the same vertical line. And if two points share the same y-coordinate, then they are horizontally separated. They lie on the same horizontal line.