PART1 Introduction to Two-Dimensional (2-D) Geometric ...

1 Two-Dimensional geometry, coordinate planegeometry, Cartesiangeometry,and planar(pronounced PLANE-er) geometry refer to thesame thing: the study of Geometric forms in the coordinate plane. Doyou remember the coordinate plane? It s a grid system in which twonumbers tell you the location of a point the first, x,tells you how farleft or right to go from the origin(the center point), and the secondnumber, y,tells you how far up or down to go. The y-axis is verticaland the x-axis ishorizontal(likethe horizon).3 Introduction toTwo- dimensional (2-D) Geometric FiguresIntroduction toTwo- dimensional (2-D) Geometric FiguresIntroduction toTwo- dimensional (2-D) Geometric FiguresPART1 2 3 4 5 7/1/04 9:12 AM Page 31 1 2 3 4 5 6 You ll see a lot more of the coor-dinate plane in geometry, butsometimes all that matters is know-ing that a figure is in the plane ortwo- dimensional without knowinga precise address for it.

2 This partwill introduce you to some of themost common figures in Two-Dimensional geometry and giveyou some names for their parts andways to work with this part, Dr. Math explains points, lines, and planes angles triangles quadrilateralsPoints, Lines, and PlanesPoints, lines, and planes correspond to talking about no dimensions,one dimension, and two dimensions in the coordinate plane. A lineis one- dimensional ,since one number, the distance from zero, tellsyou where you are. A planeis Two-Dimensional ,since you need xand yto locate a point.

3 A pointis dimensionless. It consists only oflocation, so it s only possible to be one place if you re on a point you don t need any extra numbers to tell you where you are. Points,lines, and planes are the foundations of the whole system point, line, and plane are all undefined can thatbe? Well, any definition we could give them would depend on thedefinition of some other mathematical idea that these three termshelp define. In other words, the definition would be circular!4Dr. Math Introduces 7/1/04 9:12 AM Page 4 Dear Leon,Your definition would require us to first define ray and direction.

4 Can you do that without reference to point, line, and plane ?Think of it this way: math is a huge building, in which each partis built by a logical chain of reasoning upon other parts below is the foundation? What is everything else built on?There must be some lowest level that is not based on anythingelse; otherwise, the whole thing is circular and never really startsanywhere. The undefined terms are part of that foundation, alongwith rules that tell us how to prove things are true. The goal of math-ematicians has not been to make math entirely self-contained, withno undefined terms, but to minimize the number of definitions sothat we have to accept only a few basics, and from there we will dis-cover all of math to be well defined.

5 Also, the goal is to make thoseterms obvious so that we have no trouble accepting them, eventhough we can t formally prove their put it another way, these terms do have a definition in humanterms that is, we can easily understand what they mean. They sim-ply don t have a mathematical definition in the sense of dependingonly on other previously defined terms. Dr. Math, The Math ForumIntroduction to Two-Dimensional (2-D) Geometric Figures5 Dear Dr. Math,I know that they call point, line, andplane the undefined terms of geometry, but is there a way to give those terms adefinition?

6 I ve been thinking, could aline be defined as the joining of two raysgoing in separate directions? I ve neverreally thought that anything couldn t havea definition, so is it possible for any ofthese Geometric terms to be defined?Yours truly, 7/1/04 9:12 AM Page 56Dr. Math Introduces GeometryDear Lorraine,The word point is undefined in geometry. But it is pretty easy forus to describe a point, even though it can t be defined. A point is anentity that has only one characteristic: its position. A point has nosize, color, smell, or feel. When we talk about points, we are referringto one specific example, along a number line the number 2 exists at just one point.

7 Points are infinitely small, which means the point at 2 is different from the point at Here s a picture of anumber line:If you want to distinguish one place along a number line, you point at it. You label that place with the corresponding numberand refer to it with that , how do you distinguish a location in Two-Dimensional spaceDear Dr. Math,Define a point, truly,LorraineWhat Is aPoint? 7/1/04 9:12 AM Page 6( , a sheet of paper)? Imagine that we have two number lines: onehorizontal and the other vertical. We are pointing at a place p:How do we describe where the point pis?

8 We can t just say pisat 2 because we don t know which number line that refers to. Is it at2 along the horizontal number line or the vertical one?To describe where pis, you must talk about where it is both hor-izontally andvertically. So, you can saypis at 2 horizontally and 1 verticallyHowever, this is a mouthful. Because describing points in twodimensions is really useful, we have defined some conventions tomake life easier. We call the horizontal number line the x-axis andthe vertical number line the y-axis. The convention for talking aboutpoints in two dimensions is to write(position along x-axis, position along y-axis)Therefore,pis at (2, 1)Points in two dimensions can be described by any pair of num-bers.

9 For example, (4, 5), ( , ), and ( 12, 4) are all points. Dr. Math, The Math ForumIntroduction to Two-Dimensional (2-D) Geometric 7/1/04 9:12 AM Page 7 Dear Leon,In geometry, you can think of a linejust like a normal straight line,with a couple of special features. The things that make a line ingeometry different from a line in any other context for example, artclass are that it goes on forever in both directions, it s perfectlystraight, and it s not say that their lines have zero thickness, whichis pretty hard to imagine. When we draw lines on paper, they alwayshave at least a little bit of width.

10 But when we study lines in geom-etry, we think of them as having no width at s how a lot of people draw lines on paper. The arrows at theends mean that the line continues forever in both directions:Rays and line segments are a lot like lines. A rayis like a line,except that it only goes on forever in one direction. So it starts at onepoint and goes on forever in some direction. You can think of the lightcoming from the sun as an example of a ray: the starting point is at thesun, and the light goes on forever away from the s how we draw rays:A line segmentis a little chunk of a line.