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Math Expert V
Joined: 02 Sep 2009
Posts: 59587
In the rectangular coordinate system above, if ΔOPQ and ΔQRS have equa  [#permalink]

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1
13 00:00

Difficulty:   15% (low)

Question Stats: 81% (01:19) correct 19% (01:19) wrong based on 563 sessions

### HideShow timer Statistics 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.

Attachment: 2016-10-06_1616.png [ 13.34 KiB | Viewed 6634 times ]

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Marshall & McDonough Moderator D
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Posts: 1682
Location: India
Re: In the rectangular coordinate system above, if ΔOPQ and ΔQRS have equa  [#permalink]

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7
1
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

##### General Discussion
Intern  Joined: 02 Sep 2016
Posts: 41
Re: In the rectangular coordinate system above, if ΔOPQ and ΔQRS have equa  [#permalink]

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Vyshak wrote:
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

Agree with Vyshak's explanation. Answer must be C.
Senior Manager  Joined: 06 Jun 2016
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Location: India
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Re: In the rectangular coordinate system above, if ΔOPQ and ΔQRS have equa  [#permalink]

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Bunuel wrote: 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.

Attachment:
2016-10-06_1616.png

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
Manager  S
Joined: 21 Jul 2014
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Re: In the rectangular coordinate system above, if ΔOPQ and ΔQRS have equa  [#permalink]

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Bunuel wrote: 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.

Attachment:
2016-10-06_1616.png

Hi I request you to please solve this question. Given below the solution, I couldn't get how op^2 = QS^2
Math Expert V
Joined: 02 Sep 2009
Posts: 59587
Re: In the rectangular coordinate system above, if ΔOPQ and ΔQRS have equa  [#permalink]

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2
2
rajatbanik wrote:
Bunuel wrote: 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.

Attachment:
2016-10-06_1616.png

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

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Re: In the rectangular coordinate system above, if ΔOPQ and ΔQRS have equa  [#permalink]

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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?
Math Expert V
Joined: 02 Sep 2009
Posts: 59587
In the rectangular coordinate system above, if ΔOPQ and ΔQRS have equa  [#permalink]

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\sqrt{}
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?

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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Re: In the rectangular coordinate system above, if ΔOPQ and ΔQRS have equa  [#permalink]

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_________________ Re: In the rectangular coordinate system above, if ΔOPQ and ΔQRS have equa   [#permalink] 28 Nov 2019, 04:43
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