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.

Comments:
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.