Geometry Problems of the Day (Geometry Regents, January 2014)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Geometry Regents, January 2014

Part IV: Each correct answer will receive 6 credits. Partial credit is possible.


38. In the diagram of circle O below, diameter RS, chord AS, tangent TS, and secant TAR are drawn.
[See images

below]

Complete the following proof to show (RS)² = RA • RT

Answer:

For once, the Regents was looking for one specific proof. All the steps are provided and you only have to supply the reasons you can make these statements. It's not unusual to have to write the entire proof on your own.

Notice that there are six blanks. Each reason you give will be worth one or zero credits.

Let's go through all the statements one at a time.

The first line in any proof should be all the Given information. You won't necessarily get credit for listing it (when you have to write it) but you will definitely lose credit if you do not.

RS ⊥ TS: A tangent to a circle is perpendicular to the radius it intersects.

∠RST is a right angle: This is true of all pairs of perpendicular lines.

∠RAS is a right angle: An inscribed angle is half the size of the arc it intercepts. Half of 180 degrees (the diameter intercepts a semicircle) is 90 degrees.

∠RST ≅ ∠RAS: All right angles are congruent.

∠R ≅ ∠R: Reflexive Property. This shows up in SOOOOO many proofs because it's "obvious" and they don't have to state anything for you to get it. But you have to see it and state it.

Triangle RST ≅ Triangle RAS: AA Theorem. Two angles in each triangle are congruent.

RS/RA = RT/RS: The corresponding sides of similar triangles are proportional.

(RS)² = RA • RT: Cross-multiplication. The product of the means = the product of the extremes. In the sample answers, just writing "multiplication" was ruled incorrect. I don't know if there would be an issue with just "cross-multiplication".

The last line of ANY proof should be the thing that you are trying to prove and the final Reason.

End of exam.

More to come. Comments and questions welcome.

More Regents problems.

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