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Geometry of the plane: points, segments, lines

Geometry of the plane: points, segments, lines
This week, we’ll be studying geometry, geometry of the plane. More precisely, the connection between geometrical objects, curves, for example, and lines and numbers and equations that connect the coordinates of points that are on these objects. This subject, this type of geometry is called analytic geometry, a beautiful classical subject that has many real world applications as we’ll see. So let’s begin with the plane. There it is. And we equip the plane with a set of Cartesian coordinate axes– the x-axis horizontally, the y-axis vertically and pointing up. Here’s a typical point, P, on the plane. We give it the name P for convenience to refer to it. But we can be a little more precise than that.
As we know, the position of P is described by two coordinates– its x1 coordinate horizontally, its y1 coordinate vertically. And we often reflect this fact in the notation for the point P by putting in parentheses, after the name of the point, the coordinates, x1, y1. Now, along with the point P let’s consider another given point, Q, whose coordinates are x2, y2. It’s simple to see geometrically what we mean by this segment PQ. It’s the part of the straight line determined by P and Q that’s between them. So just the part between P and Q, a line segment this is called. It’s easy to see geometrically what we mean by the midpoint of PQ. It’s the point right in the middle.
Let’s call it M. Now, how would we ever find the point M in a specific case? We could approximate it graphically, of course. But the point I’m trying to make here is that we can use numbers with our geometry to find out things. It turns out that the coordinates of M are those coordinates which are the averages of the x and y-coordinates of the two points P and Q. So the x-coordinate of the midpoint, for example, is x1 plus x2 over 2. This enables us to actually calculate it in a specific case. More notation– when we say 1/2 times a coordinate pair, x1, y1, we mean by that each coordinate is multiplied by 1/2.
And then we allow ourselves to add something similar 1/2 x2, y2. And that gives us M. So it’s a notational convention about multiplying pairs and adding them. It turns out that the segment PQ can be described very precisely at the set of all points that can be written in the following fashion. 1 minus t times the coordinate pair of P plus t times the coordinate pair of Q, where the t has to be between 0 and 1. So as you take the set of all such points letting the t vary between 0 and 1, you describe exactly the points on the segment PQ. That is, the coordinates of those points.
For example, when t is 0 in the formula, you can see that you get the point x1, y1, which is P. When t is 1, you get the point x2, y2, which is Q. And when t is 1/2, you wind up getting the midpoint that we’ve called M. Here’s an example of how to use this fact. We want to find the point whose coordinates are of the form minus 1, y and that lies on a certain segment between two given points, minus 3, 3 and 0, 5. How do we do that?
Well, if it lies on the segment, then we know that the given point has to be of the form 1 minus t times the coordinates of the first point plus t times the coordinates of the second, where the t should be between 0 and 1. We use this equation between coordinates to first group the coordinates on the right hand side and then to isolate, let’s say, the y-coordinate or the x-coordinate. Let’s do the x first. We see that 3t minus 3, the x-coordinate on the right, has to equal minus 1. This leads to the value of t. t will be 2/3. Once we know 2/3, we can now look at the second coordinate, which is y.
And it’s equal to 2t plus 3. We insert the value of t we found. We find 13/3. We found our point. Incidentally, if, in this kind of exercise, the t that we found wasn’t between 0 and 1, it would mean that the point was not in the segment in question. Now, let’s return to our plane and our two points, P and Q. Let’s introduce a third point, that I’ll call R, in such a manner that a triangle is formed, the triangle PQR. A median of a triangle means a straight line that is drawn from one of the points, let’s say P, to the midpoint of the side opposite that point P.
It is a beautiful theorem in Euclidean geometry that the intersection of all three medians is a point. They meet at a single point, which here I’ll call G. G is called the centroid of the triangle. It has other names depending on the type of application we’re doing. It can be called the centre of gravity or the barycenter at times. How do you find the centroid of a triangle? Well, numerically you do it by averaging the coordinates of the three vertices of the triangle. That is, it turns out if you take 1/3 of x1, y1 plus 1/3 of x2, y2, and so on for the third point, you wind up getting the coordinates of the centroid G.
More proof of how working with numbers gives you geometrical facts. Now, let’s look at lines. Give ourselves a line. In the first instance, we’ll take a non vertical line. So there it is. If it’s non-vertical, it must intercept the y-axis. Let’s call the coordinates of the point where that happens 0, q. 0 because x must be 0 along the y-axis, of course. Now, let’s take a horizontal line through the point 0, q. And let’s call theta the angle that is formed in this fashion. Consider now any point x, y on the line. Then a certain triangle is formed between that point 0, q and the angle theta that was formed.
Then we see that in this triangle the base of the triangle has length x and the height of the triangle is y minus q. And if you recall the definition of tangent, you see that the tangent of the angle theta is y minus q over x, opposite side over adjacent side. Solving for x, we find that y is tangent of theta x plus q. And tangent theta, therefore, is revealed to be, what we had called earlier, the slope of this straight line or affine function. We had often called it M. And q is then just the y-intercept.
The conclusion here, I wish to retain, is that if you have a non-vertical line in the plane, then that corresponds to the graph of an affine function x maps to m x plus Q. To put this another way, a point x, y will be on the line if and only if a certain equation is satisfied. Namely, m x minus y plus q equals 0. Vertical line, the other case, corresponds to an equation of the form x equals c. x minus c equals 0.
So in either case, we can summarize by saying that the general equation of a line, vertical or not, is given by the set of points whose coordinates are x, y and where the coordinates satisfy an equation a x plus b y plus c equals 0. Here, the a and the b are not both 0. So we retain that as describing a straight line by means of an equation. In fact, we’ve proved here that a straight line is the level curve of a certain affine function that is linear in the two variables x and y. Now, it’s true that a straight line is not very curvy, but we’ll see some curvier level curves later on.
We can deduce some facts about lines just from the equations that determine them. For example, if you have two lines determined by two equations linear in x y as we’ve seen, then it turns out that two lines are parallel if and only if a simple numerical criterion is satisfied between the ai’s and the bi’s that describe the lines. We can see why this is so because it turns out that either both bi’s are 0, in which case the two lines are vertical. And so they are certainly parallel. Or else neither is 0. And in that case, we can divide across in the arithmetical relationship.
And we see that the two lines have the same slope, which of course, corresponds to the lines being parallel. In a somewhat similar fashion, we can also give a criterion for two lines to be perpendicular. Namely, this one. To interpret that a little more closely, it turns out that two lines are perpendicular if and only if well, either one is vertical and 1 is horizontal, in which case they are certainly perpendicular. And in the remaining case, then the bi’s will not be 0. We can divide across. And we can get the conclusion that the product of the two slopes is minus 1.
So roughly speaking, what we’re saying here is that two lines are parallel if their slopes are the same, as the case above. And two lines are perpendicular if the product of their slopes is minus 1, a useful fact in certain geometrical problems. We’ll be turning next to circles by the way.

In this video, Francis introduces the general equation of a line in the plane. In particular conditions for two lines being parallel or perpendicular are discussed.

You can access a copy of the slides used in the video in the PDF file at the bottom of this step.

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Advanced Precalculus: Geometry, Trigonometry and Exponentials

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