PQ and RS are two chords of a circle and the lines RP and SQ

Question

1.jpg PQ and RS are two chords of a circle and the lines RP and SQ meet at O. Is OP=OQ? (1) PQ is parallel to RS (2) PR = QS

mau5 · Accepted Answer

In the given image,when the chords PR and QS are extended to meet at an external common point(O), then the following property always holds good:

OP*OR = OQ*OS

Also, this theorem gives way to the better known fact that 2 tangents from a common external point to a circle are of the same length

Back to the problem :

From F.S 1, we know that in similar triangles OPQ and OPS,  \frac{OP}{OR} = \frac{OQ}{OS}

Multiplying the above equation with the first equation, we get:  OP^2 = OQ^2 	o OP=OQ . Sufficient.

From F.S 2, we know that PR=QS. Thus,  OP*(OP+PR) = OQ*(OQ+QS) = OP^2-OQ^2+PR(OP-OQ)=0

Thus, (OP-OQ)(OP+OQ+PR)=0 . As distance can never be negative,we have OP=OQ.Sufficient.

D.

mikemcgarry · Answer

Dear udso , I'm happy to help. :-)

This is a very tricky problem. Here are a few relevant blogs: https://magoosh.com/gmat/2012/gmat-geome ... nd-angles/ https://magoosh.com/gmat/2012/inscribed- ... -the-gmat/ Statement #1 tells us the lines are parallel. Parallel lines intersect equal arcs, so arc PR = arc QS. (BTW, this also tells us the chords PR = QS). Statement #2 tells us the chords PR = QS. Equal chords intersect equal arcs, so again, arc PR = arc QS. From either of those, we know arc PR + arc PQ = arc QS + arc PQ arc RPQ = arc PQS Since these arcs are congruent, the angles that intersect them are congruent --- angle R = angle S. (Those angles are called "inscribed angles" in geometry.) Then, by the Isosceles Triangle Theorem, https://magoosh.com/gmat/2012/isosceles- ... -the-gmat/ side OR = side OS and triangle ORS is isosceles. From the two equal legs, subtract equal chords PR = QS, and we get that OP = OQ. Thus, the two statements imply each other, and either one is sufficient to deduce that OP = OQ. Answer = (D) To solve this, you would have to have quite a bit of geometry at your fingertips. This strikes me as a bit harder than what the GMAT would ask about geometry, except possibly on a very hard problem. Does all this make sense? Mike :-)

Stoneface · Answer

This doesn't seem right to me. How can statement one lead to absolute certainty? For example, PQ and RS may be parallel, but that doesn't mean they are horizontal. What if Q sat lower on the y axis than P and R sat lower than S, allowing for OR to be shorter than OS?

mikemcgarry · Answer

Dear Stoneface, The geometric truths here have nothing to do with whether any of the segments are horizontal or vertical. The ideas of horizontal & vertical are important in coordinate geometry, in the x-y plane, but in pure geometry, Euclidean geometry, the ideas of horizontal and vertical are essentially meaningless. They have absolutely no meaningful connection to any of the important idea of Euclidean geometry. If we tilt the diagram, all the geometric truths remain. tilted diagram.JPG Since PQ is parallel to RS, the two arcs, arc PR and arc QS, have to be congruent. This means that arc RPQ must be congruent to arc PQS. Angles S and angle R are the two inscribed angles that subtend these arcs, so those two angles must be congruent. Angle S = angle R. Once we establish this, then the Isosceles Triangle Theorem guarantees that OR = OS. Subtract the two equal chords, PR = QS, and we get OP = OQ. Those are the geometric relationships in the diagram, and whether the segments are horizontal is 100% irrelevant. Does all this make sense? Mike :-)

WholeLottaLove · Answer

PQ and RS are two chords of a circle and the lines RP and SQ meet at O. Is OP=OQ?

(1) PQ is parallel to RS

PQ is parallel to RS: I guess the way I look at it, is if two chords are parallel, they must form an isosceles trapezoid. If you were to increase the length of either chord they would no longer be parallel to one another - the chords angle relative to the circle would have to change to accommodate the longer or shorter line. Because they form this isosceles trapezoid, their base angles are equal to one another which means that the angles at which RP and SQ ascend are the same. This means that they will meet as a point directly above the midpoint of PQ and RS. OP=OQ. Sufficient.

(2) PR = QS

Because these lines are within the confines of a circle, there are limits in how they can be placed. If PR and QS are equal in length, then PQ and RS have to be parallel to one another. If they weren't (say, PQ was angled down relative to RS) then S wouldn't be able to intersect with Q. As with #1, we are left with PRSQ being an isosceles trapezoid meaning that OP and QO, which intersect at O, will be of the same length. Sufficient.

D

MathRevolution · Answer

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

PQ and RS are two chords of a circle and the lines RP and SQ meet at O. Is OP=OQ?

(1) PQ is parallel to RS
(2) PR = QS
 
We get OP=OQ and the question is asking whether PQ is parallel to RS if we modify the original condition, and whether OR=OS.
For condition 1, this is sufficient.
For condition 2, this is sufficient as well,
making the answer (D).

Once we modify the original condition and the question according to the variable approach method 1, we can solve approximately 30% of DS questions.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

PQ and RS are two chords of a circle and the lines RP and SQ

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

PQ and RS are two chords of a circle and the lines RP and SQ

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards