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

Question

In the diagram below, PQ is a diameter of the circle having center at O. What is the measure of ∠PTQ?

(1) ∠ROS = 40◦.
(2) ∠RPO = 55◦.

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2017-07-24_1017.png [ 12.91 KiB | Viewed 6350 times ]

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Official Answer

A

broall · Answer

2017-07-24_1017.png We have $\widehat{POS}=180^o -2\widehat{SPO}$ Also $\widehat{POS}=\widehat{POR}+\widehat{ROS}$ We also have $\widehat{POR} = 180^o -2\widehat{RPO}$ Hence $\widehat{POS}=\widehat{POR}+\widehat{ROS} \ \implies 180^o -2\widehat{SPO} = 180^o -2\widehat{RPO} +\widehat{ROS} \ \implies \widehat{ROS} = 2\widehat{RPO} - 2\widehat{SPO} \ \implies \widehat{ROS} = 2\widehat{RPS}$ (1) If $\widehat{ROS} = 40^o \implies \widehat{RPS} = 20^o$ Since $\widehat{PST} = 90^o \implies \widehat{PTS} = 90^o - \widehat{RPS} = 70^o$. Sufficient. (2) We can't know the value of $\widehat{PQT}$ so we can't know the value of $\widehat{PTS}$. Insufficient. The answer is A.

castiel · Answer

Is there any shorter way to solve this?

fskilnik · Answer

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.

https://preview.ibb.co/e2CZQq/25-Oct18-4w.gif

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

Regards, 
Fabio.

jorgetomas9 · Answer

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

fskilnik · Answer

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.

castiel · Answer

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

fskilnik · Answer

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.

castiel · Answer

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

gmexamtaker1 · Answer

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

SanjaySrini · Answer

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

fskilnik · Answer

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!

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.

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

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

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