AMC is coming up

It's a scary moment. If I screw it up, then it basically means I've wasted the past year. So this weekend has been reserved for review and AMC problems. But I couldn't live with myself if I didn't keep in contact with you people....>.>......so here's one I did recently:

Let be a trapezoid with and . Bisectors of and meet at , and bisectors of and meet at . What is the area of hexagon ? (AMC 12B 2008)

This one actually annoyed me, but I think that if you can learn how to do it correctly, then it'll help tremendously with cutting down on mistakes and such. It requires a good amount of calculations. Lots of Pythagorean Theorem with square roots. Throw that on top of the 2008 AMC being a no calculator exam, and you've got yourself a lot of paper work. If you're reading this and would actually like to learn from it, I recommend that you go through the entire problem in a very neat manner.

Anyway, the first thing is to draw a diagram. I don't have time to draw a digital one for you (sorry!), so draw it yourself =P. Here is where drawing a good diagram is crucial. Granted, it doesn't have to be perfect. In fact, all you need to do here to guarantee that you see what needs to be seen is that you draw AB and CD parallel, and draw good estimations of the angle bisectors. But if you don't, then you might miss the important observation that blows the problem away.

So, if it wasn't obvious, it should pop out that AP is perpendicular to DP, and the same with the angle bisectors from B and C. If this were true, then it would be an exercise in calculation to bash out the math leading to the answer. The steps would look something as follows:

Find the length of the height of the trapezoid.
Draw heights from A and B, and find how far the foot of each is from C and D.
Extend AP and BQ to make triangles on each side of the trapezoid.
Using the idea that the angle bisectors are perpendicular to each other, draw in some right triangles (actually, by this point, all you need should already be drawn in), and bash out the Pythagorean Theorem until you find enough information to find the areas of the triangles that are cut out of the trapezoid.

So if we can prove our conjecture, then we'll have basically solved the problem. It turns out that since A+D = 180 (AB is parallel to CD), then 1/2 A + 1/2 B = 90. Marking these angles, it shows that the angle of intersection between the bisectors is a right angle. If you were confident in your conjecture, or just didn't know how to prove it, then you could plow ahead with the problem anyway, as long as the calculations don't throw you off.

I'm not going to go through all the math for it, but I'm going to list out the "categories" that people might fall into when attempting this problem:

1) Looked, and didn't attempt:

Only do this if you're slow and want to guarantee that you get, say, the first 20 right. A good strategy for making AIME, but not really a good strategy for making USAMO. I'm going to assume that you want to make USAMO and won't have too much trouble with the first 20.

2)Looked, drew a diagram, and didn't know where to go with it:

Either your diagram was crap (it only takes 20 seconds to draw a decent diagram!), or your strategy for attacking is was poor. With geometry problems, it's important to use all the information that they give you. Something that I like to do is take what they give me, and for each one, list at least one thing that it would imply, even if it's obvious. It helps to get you going if you're stuck. Here, if you draw a decent diagram, you may not have even had to do this.

3)Looked, drew a diagram, conjectured / proved that the bisectors are perpendicular, and then didn't know where to go:

I'm guessing that this is where most 100-120 scorers would fall, which is unfortunate, because at this point, the hard part is done! They may draw in some altitudes and find some more lengths, but then get lost in their own work. Here, organization is especially important. Some tips are:

For a messy problem like this, keep your diagram seperate from your work. When calculating certain parts of the diagram, keep your work clean, write it as if you were going to hand it in, and when you get to it, box the final answer, along with what value it is. This makes it easy to refer to for later calculations. Some might argue that this takes too much time, but I usually find myself with left over time at the end if I can't get some of the problems. Don't waste time at the end, put it to good use during the exam.

When labeling your diagram, do so only if it makes the next step easier to visualize. For example, you have a number of connected right triangles, and you are going to do a Pythagorean combo, where you move from one triangle to another. Otherwise, it's just taking up room and making the diagram harder to read.

Don't get too happy drawing in extra lines. Only draw them if you have reason to believe that they might help. Example: drawing in a certain altitude that you know how to calculate, and which forms another right triangle with a side length that you need to find.

I don't really have anything else to say about this problem. It's a good excercise in work habits and basic problem solving. There might be a more clever way to do it in less time. However, assuming that you noticed the bisectors were perpendicular within 1 minute of drawing the diagram, then the rest of the calculations shouldn't take you more than 5-10 minutes. Hopefully you didn't take more than 3 minutes on any of the earlier ones =P.

Anyway, I'm doing pretty good. Second semester has been easy so far, I've taken up chess again and restarted Chess Club at my school, and I finally got a hair straightener. Good stuff. Here's some music that I've been listening to: