Some tips on getting started with geometry questions
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"Do you have any suggestions for someone who is not so good at the geometrical problems and wants to improve those skills? I guess I am a bit less visual but would love to be able to improve that area of my thinking."
Someone very thoughtfully asked this question over email. I'm not an expert at doing geometry problems, but these are just a collection of tips that have helped me along the way. Hopefully they might be useful to someone here on the forum. Whether you consider yourself good at geometry or bad at it, I'm sure you all have some advice to give for how to tackle geometry problems. Please share your tips and suggestions about how to approach geometry problems! Thank you so much!
1.) Draw a rough draft (perhaps many of them), before drawing the main diagram
Trying to draw a perfect diagram on the first go can be challenging and even distract you from thinking about the key ideas of the problem. Sometimes taking things one step at a time can help you absorb the information in the problem better. For example, consider a question that goes like this: "Parallelogram ABCD has point O as the intersection of its diagonals, which are AC and BD. Angles CAB and DBC are each twice as large as DBA, and angle ACB is r times as large as angle AOB. Find the greatest integer that does not exceed 1000r."
There is a lot of information all at once here, so focusing on only one aspect of the problem (CAB and DBC are each twice as large as DBA) helps with a rough drawing. Here we are experimenting with several rough outlines:
Focusing on one layer of the problem at a time and building up to the full complexity can help prevent you from feeling overwhelmed by all of the information!
2.) Use a ruler
Crooked lines can be deceiving. Just like how messy handwriting can spoil a computation, a geometry diagram with crooked lines can fool you into not seeing things that you should!
3.) Even better: draw a very precise diagram!
It's not so much that you'll be able to "see" the answer to the problem, but that you won't be misled by the lengths in a wrong diagram. That angle looks like a right angle, but is it? That point looks like it's where three lines intersect, but is it only two lines intersecting there? Is that two bent lines, or just one straight line? A diagram with accurate lengths and angles will help you not get deceived into thinking the wrong thing.
4.) Draw a simplified version of the problem
We did this in studying the solar panels in the Day 6 Your Turn question above! Technically, I should have drawn spheres instead of cubes, as the original problem instructs us to, but sometimes it's easier to see the area ratios with a cube. The trick is in knowing which details can be dispensed of and which are integral to the problem. This can take a bit of practice; self-reflecting on provided solutions can help.
5.) Color-code similar angles (especially with overlapping features like overlapping lines, angles, etc.)
To use the messy handwriting example again, color-coding diagrams is a way to organize the information in a problem. It's easy to end up with a cluttered diagram after you've labeled all the lengths, angles, vertices, and stuff like that. When we use colors, we don't have to write anything. It reduces the number of letters and text crowding our diagram. Imagine if instead of the pink and green colors below, we had written "p" and "g."
It's also easier to see the similar triangles in this diagram if the green and pink angles are color-coded. (The similar triangles have one obtuse angle, one pink angle, and one green angle.)
6.) Ask "What info am I not using?"
In the question above, we are trying to find the size of the angle "x." I struggled for a long time without realizing that there was some info I wasn't using: the line on the left passes through the center of the circle! With this knowledge, the problem suddenly becomes much, much easier. Try to look for this "hidden" or implicit information; it's very helpful!
7.) Try to draw extreme versions of the diagram in order to isolate the pivotal info
Sometimes you can change the angles and directions of lines in a diagram without violating the conditions. For example, in the problem below, we are trying to find the size of the angle "x" in the center of the circle.
What's the special information here? Well, there are tangent lines (lines that touch the circle at only one point). But interestingly enough, you can change the direction of the short tangent line without changing the essential question. Here are several renditions of the same question, with the pink line at different angles.
Sometimes purposefully trying to draw the diagram in a different way, while staying true to the requirements of the shapes, can help you to "see" what you need to know in order to solve the problem.
8.) Make variables for everything
I know I recommended against cluttering your diagram, but variables in general are very helpful little guys! And if you start out with tons of variables for different quantities (lengths of lines, measures of angles, etc.) you might be able to reduce them down to just a few variables as you figure out more and more relationships. This ties into the next tips, which is...
9.) Use algebra
Geometry can turn into a game of "I have lots of equations and lots of variables; which ones do I use, and in which way?" In this sense, the more equations you can come up with, the more arsenal you have at your disposal in order to attack the problem! Being really organized can really help you to keep track of your equations.
10.) Use other people's ideas
Other people's ideas, as in, in the form of theorems! In Module 2, Geometry Tools, you'll encounter some theorems, like Power of a Point, Inscribed Angle Theorem, Pythagorean Theorem, and Heron's Formula. You can think of these as shortcuts that help you get where you want to go, faster! Memorizing theorems will make it easier to go through any problem and be able to exhaust all of the possible strategies.
11.) If you have similar shapes, draw them all out in a "synchronized swimming" style
Remember how I mentioned that there are similar triangles in the diagram below? Well, they are a bit tough to see, aren't they?
