Coordinate bashing is a wonderfully underrated tool in geometry. Not only does it require much less "cleverness", it reduces complicated geometry to simple algebra. It has applications at the lowest levels up until Olympiad problems. It gets a lot of unwarranted hate for being "bashy" (which it is) and "unoriginal" (which it is). But it makes many esoteric geometry problems accessible to competitors who aren't as strong at geometry (like myself).
Simply put, it takes geometrical figures and places them into the Cartesian plane, assigning x and y values to points and defining lines as linear equations.
If you didn't pick this up from the intro, it makes complicated geometry constructions and such much simpler. There exist several simple formulae that reduce these complicated solutions into simple algebra.
There's no straight definitive answer to that. But certain aspects of a problem should make it obvious, such as:
Right triangles: Pretty much any problem with right triangles is a good choice for coordinate bashing. They translate naturally into the coordinate plane because the x and y axes form a right angle. It's almost always best to make the origin the right angle and have points
and
.
"Nice" Angles: Pretty much any angle with rational sine and cosine values (and thus rational tangent values). Rational tangent means rational slope, which means nicer values in the coordinate plane.Then you can define the angle based on the corresponding right triangle, though often with part of it missing (I'll show this in my eventual post of examples).
One or two circles: Problems with one or two circles translate well into the coordinate planes as well, especially with one centered at the origin.
Well-known plane curves: Parabolas, ellipses, etc. Because we know these by their equations in the coordinate plane, solving with coordinates is a good idea.
Problem given with coordinates: The painfully obvious case. If they give you coordinates already, then naturally coordinates is almost the best solution.
There are other applications (which I might make a post about eventually), but without calculus/integrals, this is for the most part what you can do with it. Some other cases are feasible (equilateral triangles come to mind), but these are the main cases.
There are several formulae vital to any successful coordinate bash. Here are the most important:
Slope
This one is probably the most useful (note: I'm not including the distance formula because... yeah.). Applies to any problem involving angles easily and many more.
where A is a 2x2 matrix of pairs of consecutive coordinates. Useful for finding area of strange triangles or other figures (often when they ask for the area with intersections or something)
, where the line is in the form
and the point is
. Very useful for finding lengths of different segments in your diagram. This can also be extended to the distance from a point to a plane in 3D, but 3D coordinate bashing is rarely useful (except in special cases).
This can be extended to circles not centered at the origin, but yeah, this is important.
I'll make another blog post with a bunch of examples eventually, but in conclusion, don't discount coordinate bashing as an amateur method. It really does work.
by ZK15, Apr 8, 2012, 5:03 PM 
and 
.
where A is a 2x2 matrix of pairs of consecutive coordinates. Useful for finding area of strange triangles or other figures (often when they ask for the area with intersections or something)
,
and the point is 
.
June 2012
May 2012