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

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?

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.

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

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

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.

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