Thursday, October 9, 2008

SAT/PSAT Geometry Practice: Circles and Similarity

With the PSAT rapidly approaching, here are a couple of problems which require the student to review their knowledge of circles. This will be set up as an open-ended investigation with several parts, but the content is often assessed on standardized tests. As usual there are many approaches, although efficient use of ratios and proportions is the goal here. It is critical that students thoroughly read the detailed given info in the text box.

Part I
In Figure I, determine the length of minor arc PQ.

Part II
In Figure II, determine the area of sector OPQ.

A central theme here is the relationships among the ratio of the radii of the two circles, the ratio of their intercepted arc lengths and the ratio of the areas of their corresponding sectors. Students need to have a clear understanding that one is a linear relationship and the other is a direct square variation. there are many ways to set up the solutions of these problems. We will discuss this further in the reader comments.

Also, a good discussion point is to have students explain why the last piece of given information, regarding the central angles not being congruent, was not necessary. It might be interesting to have students compute the degree or radian measures of the central angles.


Alex said...

It's a nice question, but I don't see how it's meant to help them revise circles.

Dave Marain said...

There are many important concepts and theorems relating to circles. However, I regard the relationships among the central angle, arc length and area of a sector to be fundamental. In fact, I often termed it the Basic Proportionality Theorem(s) for Circles! The ideas seem very simple, surely every geometry student knows them 'cold', or do they??

Further, the problems I posted ask students to develop a relationship between TWO circles, whereas the usual theorems relate the central angle, arc length and area of a sector of a single circle. This is the kind of applied problem typical of a slightly higher order of reasoning, typical of the PSAT/SAT. Essentially it all comes back to relationships involving similar figures (both one- and two-dimensional). A very BIG IDEA in my opinion.

Alex said...

Sorry, I suppose I struck the wrong tone. I realise that proportion and similarity are really important, not just in geometry but in many areas of maths. I was just thinking that it must be better to start a revision session on circles by looking at ideas that were specific to circles, instead of being general and powerful.

Now I've thought again, though, it might actually be beneficial to start with something the students (hopefully) know they can do, just to build their confidence up.

Dave Marain said...

No apology needed! I happen to feel strongly about ratio, proportion, part:whole relationships being one the major themes throughout K-12 math. The problems I posted in no way provide a broad review of all those 'fun' theorems about circles (angles, arcs, chords, tangents, power theorems, inscribed polygons, etc.). But the ideas in those problems happen to be fairly popular on standardized tests, the PSAT/SAT in particular. With the PSAT coming up, I wanted to reinforce this for teachers and their students.

I hope you enjoyed the problems -- they were definitely not intended to be challenges for our readers.