16.5 - Changing the Questions
Links and useful resources
- gr10's Geometry >>>START HERE<<<
- AoPS Online Textbook
- AoPS Alcumus
- Big Ideas Geometry textbook
- GeoGebra Online Geometry Constuction Tool
- Two-column math templates
- Proofs unit slides from mathgiraffe.com
- Proofs unit printables for two-column proofs
Concept summary and connections
- Look at extreme values
- Cut up and rearrange the diagram into easier pieces
When you're solving problems that have been posed to you, you can generally assume that there is a solution. That means that anything in the problem that isn't specified is free to be adjusted, as long as you respect the constraints of what was asked. It really pays off to look at the extremes in a case like that - check for things like, "What if it's zero?" or "What if they're equal?" If those things are allowed in the way the problem is set up, you can usually get some powerful clues about what to do, or even outright solve the problem, without much work at all. Consider the extremes that make things symmetrical, remove parts of the problem entirely, or simplify it in some other way.
In geometry in particular, we work with diagrams that can be very complex. Another powerful strategy is to rearrange parts of the diagram into shapes that are easier to work with. Think of an area that's broken into a bunch of pieces, but can be assembled into a square - it's much easier to compute the area of a square than to compute the areas of half a dozen odd-shaped things. If you can rearrange things without changing their sizes and shapes, and you are certain that it doesn't change the answer, then this can really help.