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Using the Cartesian Plane


Date: 01/19/97 at 04:52:15
From: Steve
Subject: geometry notation

I need to draw a rectangle that is Sqrt(2) by 1 with corners at (0,0), 
(1,Sqrt(2)), (0,Sqrt(2)) and (1,0) in the plane with lines drawn from 
(0,Sqrt(2)/2) to (1,Sqrt(2)) and from (0,Sqrt(2)/2) to (1,0).

I'm not a student but I hope you will still help.  I've never seen 
notation like this, but if you could draw it out and explain how it 
should be worked out, I would then be able to work future ones on my 
own.  I am teaching myself math and would appreciate it if you could 
tell me what level this is at so that I know how far I have to go :-)


Date: 01/19/97 at 12:01:35
From: Doctor Tim
Subject: Re: geometry notation

Hi, Steve, you brave fellow!

This is a high school level problem. Kids may encounter this type of 
thing anywhere from grade 9 to 11, depending on the program.

By "notation like this" I'm not sure which notation you mean.

Sqrt(2) means "the square root of two", the number which, when 
multiplied by itself, is 2.  That number is approximately 1.4142, but 
you can never write its decimal representation out exactly.

Two numbers in parentheses separated by a comma, like (0, 1), 
represent a point located in the Cartesian plane. (You always 
capitalize Cartesian since it's named after the French 
mathematician/philosopher Rene Descartes.)

You have to draw this to understand it:

The Cartesian plane is like an infinite sheet of paper with a 
horizontal line (the x-axis) and a vertical line (the y-axis). Where 
they intersect is called the origin. It has coordinates (0, 0). If you 
move to the right, you increase the x-coordinate, which is the first 
number. So the point on the x-axis that's one unit to the right of the 
origin is called (1, 0). One unit to the left is called (-1, 0).

The second number shows the vertical (y) coordinate, and up is 
positive. So (0, 1) is on the y-axis up one unit from the origin. 
(0, -1) is down one unit.

If you went right one unit from the origin AND up one unit (so you're 
not on an axis any more) you have to use both coordinates at once: 
(1, 1) is up and to the right of the origin.

But how far is that point from the origin? You can make a right 
triangle connecting (0, 0) and (1, 0) and (1,1) (the right angle is at 
(1, 0)).  If you know the Pythagorean theorem (capitals again!) you 
can compute it. The distance from the origin to the point (1,1) is 
Sqrt(2).

At any rate, the point (0, Sqrt(2)) that you need is up from the 
origin 1.4142... units - the same distance as the distance from (0,0) 
to (1,1). So if you have a compass, you can just match that distance 
and mark it on the y-axis.

I hope this helps. If it's just too much Greek, you may need to get a 
book or talk to somebody in person; e-mail may not be the right medium 
for this much math (what I've just described is kind of the basis for 
all of analytic geometry), especially when it's so visual.

Good luck!

-Doctor Tim,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Coordinate Plane Geometry
High School Geometry

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