17.4 - Proofs with Analytic Geometry
Links and useful resources
Concept summary and connections
- You can place a coordinate system anywhere, at any angle. Choose one that makes your life easy!
- Choosing a coordinate system that makes as many things as possible equal to zero is a good place to start
Once again, we're tying things we know from algebra together with things we know from geometry. One of the biggest hurdles is understanding how to connect them, and the first place we start is with the coordinate system. Coordinates act like translators, taking the lengths and angles of pure geometry into the realm of things we can use tools of algebra on. We're always working with an abstract idea of a shape, and we have different representations of that shape based on the various mathematical tools we have at hand. Later, you'll learn how to use even more representations. The important thing to realize is that the shape itself is what connects all of these representations. If something is true algebraically for a right triangle, it's true for any representation of a right triangle. Here's what you need to remember:
- You can put the coordinate system's origin ANYWHERE YOU WANT, and you can orient it in ANY DIRECTION! So, pick one that makes your life as easy as possible. Just make sure the axes are perpendicular, and the divisions are all the same size.
- Choosing good axes makes many things into zeros, and makes other things be equal to each other
- Using variables as the individual coordinates means that your representation is valid for all possible variations of that shape!
- Just be careful not to assume that things line up on the coordinates a certain way because the diagram looks like it should - you have to know the properties of your shape. For example, the legs of a right triangle will always align with perpendicular axes, but that does not mean all triangles will. Make sure you're not proving something about a restricted class of shapes by accident!
- It's useful to put a vertex at the origin most of the time, but always think about it before you do because there might be a better place as well.
- If you don't see a solution immediately, try a few different ways of locating your coordinate system. Look for:
- Ways to put a segment of the diagram directly on an axis
- Ways to put a vertex in an easy location
- Ways to use symmetry in combination with the origin to simplify the math
- Many people think of those tricks as cheating, because what if you can't do that some time??" None of those things is cheating, they're just the right thing to do in order to solve the problem. In other words, cheat as much as possible!
- If you come across a problem where you can't use these tricks, that just means it's a different kind of problem, and it's going to be a lot harder. So don't borrow trouble.
Guided practice
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