A step-up SAT/GRE problem -- Circles (Rhombus edition)

In an earlier post, we talked about "step-back problems." The idea is that, wherever possible, each problem should be associated with at least one problem that uses similar format and relies on similar concepts but which "steps up" (is more difficult) or "steps down" (is easier).

In that previous post we talked about problems where you had to find the shaded area of a circle. This problem covers similar territory but takes things up a notch.

Circle 1 and Circle 2 both have radius 2. Each passes through the center of the other. Find the area of the rhombus formed by the two points of intersection (A and B) and the centers of each circle (C1 and C2).

Solution after the break.

A note on creating geometric graphics

As you can see in the previous post, I've been working on a project involving the kind of problems you might see on the SAT or GRE. I've been focusing on geometry to start with which means I have to create lots of geometric drawings.

I recently started using Inkscape and I would highly recommend it for the following reasons:

It's a free, open-source program;

It works with  Scalable Vector Graphics (SVG). Scalability is a great feature;

It's easy.

Quick tip: you can draw circles and squares from a center point using the shift key.

A step-back SAT/GRE problem -- Circles

I've been thinking a lot about video learning, particularly video-assisted self-study. One of the ideas I've come up with is the step-back problem.

Sometimes, when you're trying to teach yourself a subject or study for a test like the SAT or GRE, you will run into a problem that still confuses you after you've gone through the explanation. If you have access to a live instructor,  you can always ask for a more detailed explanation but what do you do if you are trying to learn the subject from YouTube videos?

My thought is to pair up problems of medium to high difficulty with problems that use some of the same concepts but are much simpler.

Here's an example. The following is very similar to some problems you are likely to see on the SAT or GRE. Try it on your own then check below the fold for the answer. If you didn't get the right answer and still don't understand what you got wrong even after reading the explanation, try the second problem.

Circle 1

The radius of circle 1 is 5. Both line segments pass through the center of the circle. Find the area of the shaded region.

Puzzling your way through the GRE -- logic edition

I was looking through a GRE sample test the other day and I came across a question that looked oddly familiar. I am not going to repeat it verbatim (not quite sure of the copyright status) but the gist was that the question gave some names and some properties associated with those names and some conditions.

After a moment, I realized that this problem reminded me of an old-fashioned logic grid puzzle.

Here's an example from a popular site:

You can eliminate pairs you know aren't true with an X, and pencil in pairs you know are related with an O. If you know, for example, that Lauren wasn't born in 1961, you can add an X in the box where the Lauren column and 1961 row meet. Similarly, if you know that Bryant was born in 1971, you can add an O in the appropriate box. Furthermore, since every option can only be used once in any given puzzle, you can eliminate the four other options for Bryant in that category (1937, 1946, 1961, 1975) and the four other options for 1971 (Anahi, Jayden, Lauren and Nikolas).

You probably won't see a grid like this on the GRE or the SAT but you will run across the underlying concepts so if you're preparing for one of these tests, it might be worth your while to spend a little of your study time as play time with these puzzles.