In the diagram below, PQ is a diameter of the circle having center at

Statement (1) can be solved exploring two facts only:

(i) Angle PSQ (hence angle PST) is (are) right (because PQ is a diameter)

(ii) Angle RPS is inscribed with the corresponding 40-degrees central angle ROS

Statement (2) is not sufficient, as proven with the geometric bifurcation.




This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Excuse me, why 40 divided by 2?, could you explain it, thank you.

Hi, jorgetomas9 !

Thank you for your interest in my solution.

Every inscribed angle (in this case RPS) is half the value of the central angle (in this case ROS) corresponding to the same arc (in this case arc RS).

Regards,
Fabio.

That bifurcation method will come handy in various DS problems of geometry.
Thanks fskilnik.

Hi, castiel.

Thank you for the nice compliment (and for the kudos)!

You are absolutely right. Bifurcations are extremely useful and, to be honest, they change intuition (I feel...) into certainty (I am sure...)!

This "shield" is also possible in algebraic and geometric-algebraic bifurcations, by the way. (You can find them in my previous posts... and in my course.)

Regards and success in your studies,
Fabio.

I'll check them as well.
Thanks again!!

Hi Bunuel could you please provide an explanation for this problem ?

Hi, could you explain how geometric bifurcation works and how you used it to prove statement 2 is insufficient

Hi SanjaySrini ,

Thank you for your interest in our solution/method.

The "BIFURCATION" (a word we are proud to have introduced into the Data Sufficiency GMAT context, borrowed from the Ordinary Differential Equations nomenclature) is (we believe) the proper way to change "feeling" into "certainty".

I mean: you believe (say) statement (2) is not enough to the uniqueness of your FOCUS, say, the length of some line segment presented in a given geometry problem.

What do we suggest? Find two different geometric configurations, both of them satisfying all the question stem (pre-statements) and also statement (2), and each one of them giving a DIFFERENT value for your focus!

If you do that (properly), you are SURE statement (2) is not enough. As we say, you change your beliefs into mathematical truths. You are "bullet-proof", you don´t need to see the official solution, for instance!!

People will say you will not have time during your test to do that. We respect other people´s opinion/approach but... with all due respect, we believe they are wrong in (at least) two ways:

1. The process of developing your skills into proper (and quickly-built) bifurcations will develop your *mathematical maturity* and this is EXACTLY what the quantitative section is all about. In other words, your are getting stronger during your studies, even for Problem Solving questions, of course!

We like to give our students the following analogy: no boxer will use ropes during his real fights, but ALL serious boxers train using ropes, for rhythm, for balance, for breathing capacity, etc. In other words, you have to prepare for a fight using things you know you will not use in the fight itself!

[Well, you WILL bifurcate during your test, anyway, although on many occasions it won´t be needed because you will be sure you WOULD be able to bifurcate if you would like to.
Example: Is x>0? (2) x is even.
I believe you know that there are positive and negative even numbers, therefore the bifurcation would be trivial, hence not needed.
Anyway, let´s do it properly, with a special twist... we will not mention negative numbers: Take 0 for a "no", 2 for a "yes".
Now we are SURE statement (2) is not sufficient. It´s nice, isn´t it?]

2. Most of your Data Sufficiency problems in the REAL exam will be VERY similar to the ones you would have been studying with us! (To be honest, our Data Sufficiency problems are VERY hard, exactly to make you feel at ease during your test, even when performing for outstanding performance.)
Therefore if you have bifurcated something very similar before, you will be able to repeat or adapt the reasoning VERY fast and this will take much less time in the real test than while you are studying for the test!

In other words, we do NOT expect you to do things as rigorously and as thoroughly during your real GMAT, exactly because you did that before it while preparing for it!

Well, I hope you got the point.

Regards and success,
Fabio.