Math SATs- Analytic Geometry

Posted Wed, 07 May 2008 19:43:08 -0000 by Oren Lahav

Welcome to a Math SAT Preparation Lesson. In this lesson we will study the wonders of analytic geometry.

While the geometry we've discussed before is based on real-world shapes and figures, this isn't always the case. As a matter of fact, mathematicians like to take geometry and put it on a graph-paper, which leads to the concept of analytic geometry. This is basically geometry on a numeric grid.

To simplify matters, we'll only look at the real number line and at 2-D analytic geometry. If you pass your SATs and continue to study math in college or university, you'll get exposed to the horrors of multi-variables calculus and analytic geometry in multiple dimensions.

In the meanwhile, let's introduce the basics

First of all, the XY grid. It looks like this: .

Each spot on our grid can be represented by a coordinate pair- (x,y). The origin is designated by (0,0), and positive x values are on the right, negative x values are on the left, positive y values are up and negative y values are down.

Cool. Now, let's start with the basic and most important shape in analytic geometry- lines.

A line equips a linear function in terms of our x-values to the corresponding y-values. If that sentence didn't make any sense to you yet, that's ok. All it means is that a line has the formula Ax+By=C for any A, B or C. Note how the x and y both have degree 1, this is important. The general form of a line can also be written as y=mx+b , where m is the slope of the line and b is its y intercept.

Say you have 2 points, A( x_1 , y_1 ) and B( x_2 , y_2 ) and we want to draw a line between them. What would be the equation of that line? First, we find the slope of the line. This is easily done using the formula: $m=\frac{y_2 - y_1}{x_2 - x_1}$ . Now all we have to do is plug one of the points into y_1=m * x_1+b and solve for b to find our generic equation.

There are 2 special types of lines that you should think about.

Horizontal lines are parallel to the x-axis and have an equation in the form of y=c , where c is a number, like 6. These lines have slope 0. Vertical lines have a generic equation that looks like x=c , where c is again any real number. Vertical lines are said to have no slope, or a slope of infinity in some cases. This lines are a bit tricky, but don't be afraid, they're simple to handle- all a horizontal line means is that for any x value, the y value is the same. So for the line y=6 , the points (0,6), (1,6), and (-72,6) are all on the line. Vertical lines work in a similar manner.

There are a couple of neat little trick with slopes:

If I want to find the slope of a line perpendicular to the line with slope m, all I have to do is find $\frac{-1}{m}$ . For example, if the slope of my line is 2, the slope of a perpendicular line would be $\frac{-1}{2}$ .

Another thing to do with slopes is reflections. A reflection of a line on the x-axis will result in the negative value of the entire line. A reflection on the y-axis will only result in the opposite (negative) slope. For example, reflecting y=x+1 on the x-axis will give you y=-x-1 , while reflecting it on the y-axis will yield y=-x+1 .

We can do many things with lines now- for example, given the equations of 2 lines, we can solve for x and y to find their intersection. We can also do this for shapes other than lines. Shapes like this will be discussed in more details in the Basic Functions lesson.

A final thing to know is the distance equation between 2 points.

Note that this distance really calculates the length of the line connecting these points, since as we all know the shortest distance between 2 points is a straight line. This distance formula for point A( x_1 , y_1 ) and B( x_2 , y_2 ) is $d = \sqrt{ (x_1 - x_2) ^ 2 + (y_1 - y_2) ^ 2}$ .

Also, there's an easy way to find a mid-point, which is the point in the middle of the line segment, between two points. Say my points are A( x_1 , y_1 ) and B( x_2 , y_2 ), my mid point would be M( $\frac{x_1 + x_2}{2}$ , $\frac{y_1 + y_2}{2}$ ). This is an easy and quick way to find mid-points.

After you've memorized this formulas and practiced getting some shapes on a grid paper, feel free to test your knowledge with the Analytic Geometry test.

Image Credit : midom

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