In the rectangular coordinate system above, if ΔOPQ and ΔQRS have equa

In the rectangular coordinate system above, if ΔOPQ and ΔQRS have equal area, what are the coordinates of point R ?

Since ΔOPQ and ΔQRS have equal area, then \(\frac{1}{2}*OP*OQ = \frac{1}{2}*RS*QS\), which gives \(OP*OQ = RS*QS\).

(1) The coordinates of point P are (0,12). This implies that OP = 12. Not sufficient.

(2) OP = OQ and QS = RS. No values are given, so this statement is clearly insufficient. But from this statement we get that \(OP^2 = RS^2\) (from \(OP*OQ = RS*QS\)), which gives OP = RS. So, OP = OQ = QS = RS. Basically we get that ΔOPQ and ΔQRS are congruent, similar triangles.

(1)+(2) \(OP^2 = RS^2=12^2\). So, \(OP = OQ = QS = RS=12\). Thus the coordinates of R are (24, 12). Sufficient.

Answer: C.
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ΔOPQ and ΔQRS have equal area --> 0.5*OP*OQ = 0.5*QS*RS

St1: The coordinates of point P are (0,12) --> OP = 12
12*OQ = QS*RS
Clearly insufficient as we do not know the lengths of OQ, QS and RS.

St2: OP = OQ and QS = RS --> Not Sufficient as we do not know the values of the given lengths

Combining St1 and St2: OP^2 = QS^2 --> OP = QS
Hence OP = OQ = QS = RS = 12
Co-ordinates of R = (24,12)
Sufficient

Answer: C

Agree with Vyshak's explanation. Answer must be C.

Statement 1 gives only y co-ordinate value. Without knowing the distance OS we cannot find the co-ordinate of point R.
In Suff..

Statement 2 says both the triangles are isosceles but this data is not suff to find the co-ordinates.
In Suff

Combining we have x co-ordinate of R as 24 and y co-ordinate of R as 12
R (24,12)
Hence C

Hi I request you to please solve this question. Given below the solution, I couldn't get how op^2 = QS^2

For statement 1, can't a pythagorean triplet apply? That way A is also sufficient. (12,13,5)
B) is sufficient on its own too, giving us triangle 1 with (0,0) (0,12) and since two sides are equal, (12,0)for the coordinate Q.
And since we have (12,0) we get 24,0 and the last coordinate of R

Could someone explain why D is wrong? And what is the error in the above logic?

\sqrt{}

For (1) infinitely many other cases are possible. For example, (12, 1, \(\sqrt{145}\)), (12, 2, \(\sqrt{148}\)), (12, 1.5, \(\sqrt{146.25}\)), ... Generally knowing only one side of a triangle is not enough to find other sides.

For (2) no values are given. You cannot use info from one statement when solving another.
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Since ΔOPQ and ΔQRS have equal area,
then (1/2) ∗ OP ∗ OQ = (1/2) ∗ RS ∗ QS
--> OP ∗ OQ = RS ∗ QS.

Statement 1: P is (0,12) --> OP = 12 --> Insufficient.
Statement 2: OP = OQ & QS = RS --> OP2 = RS2 Still no info about R --> Insufficient.
Combining 1 & 2: OP = OQ = QS = RS = 12 --> Thus, R is (24, 12) --> Sufficient. Hence, answer C

In the rectangular coordinate system above, if ΔOPQ and ΔQRS have equal area, what are the coordinates of point R ?

(1)The coordinates of point P are (0,12).

If the coordinates of point P are (0,12), we still can't determine what the coordinates of point R are. ΔOPQ and ΔQRS have equal area, but we don't know what the sides are. INSUFFICIENT.

(2) OP = OQ and QS = RS.

Clearly insufficient. We have two right triangles, but we can't determine the coordinates of point R.

(1&2) If point P is (0, 12) then OQ = QS = SR = 12. The coordinates of point R are (24, 12). SUFFICIENT.

Answer is C.
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(1) \(\frac{1}{2}OP*OQ=\frac{1}{2}*QS*RS\); Given OP=12

We don't know about OQ, QS or RS; Insufficient

(2) No specific measurement is given. Insufficient.

Considering both:
No additional information we can derive; Insufficient.

The answer is E.

Solution:

Question Stem Analysis:

We need to determine the coordinates of point R, given that right triangles OPQ and QRS have the same area.

Statement One Alone:

Knowing only the coordinates of point P is not enough to determine the coordinates of R. Statement one alone is not sufficient.

Statement Two Alone:

We see that both triangles are right isosceles triangles (i.e., they are each 45-45-90 triangles). However, we can’t determine the coordinates of R without knowing any coordinates of vertices such as P, Q and/or S. Statement two alone is not sufficient.

Statements One and Two Together:

Since right triangle OPQ is isosceles and P = (0, 12), then Q = (12, 0). Since both right triangles are isosceles and they have the same area, OQ = QS and OP = SR. Since Q = (12, 0), then S = (24, 0) so that OQ = QS. We see that R has the same x-coordinate as S and since OP = SR, so R must have the same y-coordinate as P; therefore, R = (24, 12). Both statements together are sufficient.

Answer: C
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ScottTargetTestPrep chetan2u IanStewart Bunuel Are we assuming Q to be positive here? Why can't it be (-12,0)? Is it because Q is shown as positive in the figure given? I thought it can lie in fourth quadrant too. My understanding of GMAT questions has been that the figures given in the questions are signless, meaning they necessary not be in any quadrant.

Yes, we are assuming Q is positive here. In general, while you can't trust anything about the scale of a DS diagram, you can trust the general ordering and direction of points, and you can trust that things that look like straight lines are indeed straight lines. So here, we can assume that Q and S are on the x-axis, that the points O, Q, and S are in that order and not some other order going from left to right, that Q and S both have a positive x-coordinate, and that P is on the positive part of the y-axis. But we can't (without using the statements) conclude anything about how big the two triangles are, or how long any of the lines are, even if some of them might look the same in the diagram.
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