In the rectangular coordinate system

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In the rectangular coordinate system

by BTGmoderatorLU » Sun Jul 15, 2018 9:21 am

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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).
(2) OP = OQ and QS = RS.

The OA is C.

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?

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by GMATGuruNY » Sun Jul 15, 2018 12:22 pm

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BTGmoderatorLU wrote:Image

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).
(2) OP = OQ and QS = RS.
Statement 1:
Thus, OP=12.
Case 1: OQ=1, QS=1, and RS=12, with the result that the area of reach triangle = (1/2)(1)(12) = 6
In this case, R = (2, 12).
Case 2: OQ=1, QS=12, and RS=1, with the result that the area of reach triangle = (1/2)(1)(12) = 6
In this case, R = (13, 1).
Since R can have different coordinates, INSUFFICIENT.
INSUFFICIENT.

Statement 2:
Since both triangles are isosceles right triangles and have equal areas, all four legs must be equal:
OP=OQ=QS=RS.
Case 3: OP=OQ=QS=RS=12, with the result that the area of each triangle = (1/2)(12)(12) = 72
In this case, R = (24, 12).
Case 4: OP=OQ=QS=RS=2, with the result that the area of each triangle = (1/2)(2)(2) = 2
In this case, R = (4, 2).
Since R can have different coordinates, INSUFFICIENT.

Statements combined:
Since OP=12 and OP=OQ=QS=RS, only Case 3 is possible:
OP=OQ=QR=RS=12.
Thus, R = (24, 12).
SUFFICIENT.

The correct answer is C.
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by [email protected] » Thu Aug 30, 2018 11:32 am

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BTGmoderatorLU wrote:Image

In the rectangular coordinate system above, if triangle OPQ and triangle QRS have equal area, what are the coordinates of point R?

(1) The coordinates of point P are (0, 12).
(2) OP = OQ and QS = RS.
Our solution is almost "Algebra-free"... please check the image attached to see our GEOMETRIC BIFURCATION in action!

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


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BTGmoderatorLU wrote:
Sun Jul 15, 2018 9:21 am
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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).
(2) OP = OQ and QS = RS.

The OA is C.

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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