Subject: Trigonometry

Coordinate Geometry

So What is Coordinate Geometry?

Coordinate Geometry has a few synonyms: cartesian geometry (probably one of the more well-known identifiers), analytic geometry, or analytical geometry. So if your teacher is calling it one of these other names in class, no worries.

The Breakdown

The easiest way to remember the general meaning of Coordinate Geometry is to think of yourself as a taxicab driver. You can only take streets by the block; you can't quite cut through buildings, right? Well, to start with, that's how we'll envision ourselves to better understand coordinate geometry.

As a taxi cab driver we're going to stick to straight lines at first.. Then we'll start getting creative. To recap a little Geometry here is a coordinate plane:

As you can see a standard graph with an x and y-axis along with numbers along the x-axis.

Distance between two points:

Finding the distance between two points boils down to a simple formula. To understand the scope of this equation, understand that the entire world is on a coordinate system (longitude and latitude) and your house may be one point on that large coordinate system, and your friend Matt's may be another. If you wanted to find the distance between your house and Matt's house you could use this equation.

Or this same equation can be expanded into three dimensions and used for space travel (don't you think it's important for astronauts to know how far they're going?).

Given two points, (x1, y1) and (x2, y2), the distance between these points can be found as:

\sqrt{{(x_2 - x_1)^2} + {(y_2 - y_1)^2}}

And for all you future physicists, for three dimensions you could use: (x1, y1, z1) and (x2, y2, z2), & the distance between these points can be found as:

\sqrt{{(x_2 - x_1)^2} + {(y_2 - y_1)^2} + {(z_2 - z_1)^2}}

Examples:

Find the following Distances:

  1. Between (1,7) and (9,4)
  2. Between (-5,7) and (9,4)
  3. Hard: Between (-2,7,12) and (5,-9,14)

1) \sqrt{{(9 - 1)^2} + {(4 - 7)^2}} = 8.544
2) 14.3178
3) 16.4012

Midpoint of a line segment:

Finding the midpoint between two points is normally ever used to make word problems more difficult. But since it is used, you will have to know how it's done (but don't worry, this will be mindblowingly easy). You could figure out the length and slope of the line, convert it to a distance along the x-axis, and then figure out the y coordinate through the 'y=mx+b' equation.

Or take the short short short-cut, where given (x1, y1) and (x2, y2) the midpoint of the segment is:

\left({{x_1+x_2\over2} , {y_1+y_2\over2}}\right)

Examples:

Find the following Distances:

  1. Between (2,17) and (6,5)
  2. Between (-5,-7) and (9,-4)
  3. Between (12, -12) and (-4, -7)

1) \left({{2+6\over2} , {17+5\over2}}\right) = (4, 11)
2) (2, -5.5)
3) (4, -9.5)