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Manager  Joined: 02 Dec 2012
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In the xy-coordinate plane, is point R equidistant from  [#permalink]

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In the xy-coordinate plane, is point R equidistant from points (-3,-3) and (1,-3) ?

(1) The x-coordinate of point R is -1.
(2) Point R lies on the line y = -3.
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In the xy-coordinate plane, is point R equidistant from points (-3,-3) and (1,-3) ?

Look at the diagram below: Notice, that the green line (x=-1) is the perpendicular bisector of the line segment with endpoints (-3,-3) and (1,-3), thus ANY point on this line will be equidistant from points (-3,-3) and (1,-3).

(1) The x-coordinate of point R is -1 --> point R is on the green line. Sufficient.
(2) Point R lies on the line y = -3 --> point R may or may not be on the green line. Not sufficient.

Attachment: Equidistant points.png [ 9.68 KiB | Viewed 31190 times ]

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In the xy-coordinate plane, is point R equidistant from  [#permalink]

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We have to find out whether $$R(x,y)$$ is equidistant from the two points mentioned

Using the distance formula
$$(x+3)^2+(y+3)^2 = (x-1)^2+(y+3)^2$$
$$(x+3)^2 = (x-1)^2$$

So basically we have to prove whether $$(x+3)^2 = (x-1)^2$$or not?

1)Substituting$$-1$$ in the above equation $$(x+3)^2 = (x-1)^2$$ results in it being equal
Thus sufficient

2)$$y = -3$$wouldn't help us with this eqn:$$(x+3)^2 = (x-1)^2$$
Thus insufficient

Ans is A
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Manager  Joined: 21 Jan 2010
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Re: In the xy-coordinate plane, is point R equidistant from  [#permalink]

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In the xy-coordinate plane, is point R equidistant from points (-3,-3) and (1,-3) ?

(1) The x-coordinate of point R is -1.
(2) Point R lies on the line y = -3.

Any point that lie on the perpendicular bisector the line segment with extreme points (-3,-3) and (1,-3) will satisfy this condition. The perpendicular bisector of the line segment is x=-1.

1) this means the point lies on x=-1. Sufficient.
2) This may or may not lie in the middle. The point -1,-3 is the mid point of the line segment but their are other points on the line such as (-2,-3) which doesn't satisfy the requirements. Insufficient.

Hence A.
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Re: In the xy-coordinate plane, is point R equidistant from  [#permalink]

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Why does the green line go through point -1 ? I couldn't find any chapter which explains this system.. can anyone help?
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Re: In the xy-coordinate plane, is point R equidistant from  [#permalink]

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I got really scared seeing this question. But visualizing the coordinate plane made for a much simpler approach.

Once I got the two points mapped out it was obvious that point R had to be on X = -1.

I suppose this was possible because the two points had the same Y coordinates, which allowed for several a straight line at equidistance from the two points. Has anyone got any suggestions or Q's that involves points without this possibility? e.g. A=(1,0) B=(6,6)
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: In the xy-coordinate plane, is point R equidistant from  [#permalink]

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Hi MarkusKarl,

Most Geometry questions have a "visual" component to them, so drawing the "work" involved (the shapes, the graph, etc.) will almost always be beneficial - in that way, you can connect conceptual ideas to real-world examples. GMAT questions in general are almost all pattern-based, so if you find yourself 'stuck' conceptually, you have to think about the rules involved and simplify the logic.

In your example, you name two points that don't share an X or Y co-ordinate, but the concept involved in this prompt applies to your example as well. There WILL be a "line" of co-ordinates that are equidistant from the two points that you named (it's just that the "line" will be a diagonal line and will NOT involve any shared X or Y co-ordinates).

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Re: In the xy-coordinate plane, is point R equidistant from  [#permalink]

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Bunuel wrote:
Kchaudhary wrote:
In the xy-coordinate plane, is point R equidistant from points (-3,-3) and (1,-3)?

(1) The x-coordinate of point R is -1.
(2) Point R lies on the line y=-3.

Merging topics. Please refer to the discussion above.

Hi Bunnel, in statement 1, how can you consider that y-coordinate of R is 0?
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Re: In the xy-coordinate plane, is point R equidistant from  [#permalink]

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Kchaudhary wrote:
Bunuel wrote:
Kchaudhary wrote:
In the xy-coordinate plane, is point R equidistant from points (-3,-3) and (1,-3)?

(1) The x-coordinate of point R is -1.
(2) Point R lies on the line y=-3.

Merging topics. Please refer to the discussion above.

Hi Bunnel, in statement 1, how can you consider that y-coordinate of R is 0?

The y-coordinate of R is not necessarily 0. The point is that since x-coordinate of R is -1, then R is on the green line, so no matter what is the y-coordinate, R will be equidistant from the given points. _________________
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Re: In the xy-coordinate plane, is point R equidistant from  [#permalink]

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Hi lou34,

Thats because S1 says so.

S1 says - The X-co-ordinate of Point R is -1. This mean it will have to R WILL have to be ANY point on the green line mentioned by Bunuel.

I hope this clears your question.

lou34 wrote:
Why does the green line go through point -1 ? I couldn't find any chapter which explains this system.. can anyone help?

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Re: In the xy-coordinate plane, is point R equidistant from  [#permalink]

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HI Kchaudhary,

Just adding to Bunuel's solution.

Even though the term perpendicular bisector is used, we can infer S1 is sufficient without knowing about it. A quick sketch of the Co-ordinate will be able to get us there.

Consider the attached image. In the Image I have taken an Arbitrary point R on the green line. Named the given two points as A and B. Also labelled O for easier understanding.

What is asked is - AR = BR?. If you notice - Point A, B and R form two Right angles at common point O. In these two Right angled triangle, two sides are equal. AO = AB = 2 (Follows from given co-ordinates) and OR is common to both Right triangles. So, it clearly follows that the third side of both right angles MUST be equal. Meaning AR = BR. This makes S1 sufficient.

I hope this helped Kchaudhary wrote:
Bunuel wrote:
Kchaudhary wrote:
In the xy-coordinate plane, is point R equidistant from points (-3,-3) and (1,-3)?

(1) The x-coordinate of point R is -1.
(2) Point R lies on the line y=-3.

Merging topics. Please refer to the discussion above.

Hi Bunnel, in statement 1, how can you consider that y-coordinate of R is 0?

Attachments

File comment: Named given point A and B for clarity. Assumed Point R on green line Equidistant points.png [ 18.45 KiB | Viewed 18996 times ]

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In the xy-coordinate plane, is point R equidistant from  [#permalink]

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In the xy-coordinate plane, is point R equidistant from points (-3,-3) and (1,-3) ?

(1) The x-coordinate of point R is -1.
(2) Point R lies on the line y = -3.

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.

From the original condition, we can set up as this question as follows.
The distance between $$R(x,y)$$ and $$(-3,-3)$$ is $$\sqrt{(x+3)^2 + (y+3)^2}$$ and the distance between $$R(x,y)$$ and $$(1,-3)$$ is $$\sqrt{(x-1)^2 + (y+3)^2}$$.
Thus, we have $$\sqrt{(x+3)^2 + (y+3)^2} = \sqrt{(x-1)^2 + (y+3)^2}$$.
Then $$x^2 + 6x + 9 + y^2 + 6y + 9 = x^2 -2x + 1 + y^2 + 6y + 9$$.
$$6x + 9 = -2x + 1$$
$$8x = -8$$
$$x = -1$$

We have 2 variables $$x$$ and $$y$$ and 1 equation, $$x = -1$$.
In the original condition, there is 1 variable(x), which should match with the number of equations. So you need 1 equation. For 1) 1 equation, for 2) 1 equation, which is likely to make D the answer.

For 1), $$x = -1$$, which is equivalent to the condition from the original question. No additional condition is provided. Thus this is not sufficient.

For 2), $$y = -3$$. Then the point R is (-1,-3). This is sufficient.

Therefore, the answer is B.

-> For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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Re: In the xy-coordinate plane, is point R equidistant from  [#permalink]

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MathRevolution wrote:
In the xy-coordinate plane, is point R equidistant from points (-3,-3) and (1,-3) ?

(1) The x-coordinate of point R is -1.
(2) Point R lies on the line y = -3.

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.

From the original condition, we can set up as this question as follows.
The distance between $$R(x,y)$$ and $$(-3,-3)$$ is $$\sqrt{(x+3)^2 + (y+3)^2}$$ and the distance between $$R(x,y)$$ and $$(1,-3)$$ is $$\sqrt{(x-1)^2 + (y+3)^2}$$.
Thus, we have $$\sqrt{(x+3)^2 + (y+3)^2} = \sqrt{(x-1)^2 + (y+3)^2}$$.
Then $$x^2 + 6x + 9 + y^2 + 6y + 9 = x^2 -2x + 1 + y^2 + 6y + 9$$.
$$6x + 9 = -2x + 1$$
$$8x = -8$$
Got...methodical way.
$$x = -1$$

We have 2 variables $$x$$ and $$y$$ and 1 equation, $$x = -1$$.
In the original condition, there is 1 variable(x), which should match with the number of equations. So you need 1 equation. For 1) 1 equation, for 2) 1 equation, which is likely to make D the answer.

For 1), $$x = -1$$, which is equivalent to the condition from the original question. No additional condition is provided. Thus this is not sufficient.

For 2), $$y = -3$$. Then the point R is (-1,-3). This is sufficient.

Therefore, the answer is B.

-> For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
Got it..methodical way.
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In the xy-coordinate plane, is point R equidistant from points (-3,-3) and (1,-3) ?

(1) The x-coordinate of point R is -1.
(2) Point R lies on the line y = -3.

Target question: Is point R equidistant from points (-3,-3) and (1,-3)?
This question is a great candidate for rephrasing the target question.

First sketch the two given points Notice that the point (-1, -3) is equidistant from the two given points. MORE IMPORTANTLY, every point on the line x = -1 is equidistant from the two given points. So, we can rephrase the target question . . .
REPHRASED target question: Is point R on the line x = -1?

Statement 1: The x coordinate of point R is -1
If the x-coordinate is -1, then point R is definitely on the line x = -1
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: Point R lies on the line y= -3
This tells us nothing about whether or not point R is on the line x = -1?
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

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Re: In the xy-coordinate plane, is point R equidistant from  [#permalink]

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In the xy-coordinate plane, is point R equidistant from points (-3,-3) and (1,-3) ?

(1) The x-coordinate of point R is -1.
(2) Point R lies on the line y = -3.

Key in this question is paying careful attention to the wording: the Q focuses on equidistant however the initially urge is to interpret this as midpoint (which is incorrect).

Since both points have the same y-values, any point that is half-way between their x-values will be equidistant. Hence (1) is sufficient since it tells us what the x-value is. (2) is insufficient alone since it tells us nothing about the x-value. Answer is A.
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Re: In the xy-coordinate plane, is point R equidistant from  [#permalink]

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Bunuel wrote:
In the xy-coordinate plane, is point R equidistant from points (-3,-3) and (1,-3) ?

Look at the diagram below: Notice, that the green line (x=-1) is the perpendicular bisector of the line segment with endpoints (-3,-3) and (1,-3), thus ANY point on this line will be equidistant from points (-3,-3) and (1,-3).

(1) The x-coordinate of point R is -1 --> point R is on the green line. Sufficient.
(2) Point R lies on the line y = -3 --> point R may or may not be on the green line. Not sufficient.

Attachment:
Equidistant points.png

Here in Statement(2),Point R lies on the line y = -3 then the only point on this line y=-3 equidistant from points is (-1,-3),.....so why not D?
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Re: In the xy-coordinate plane, is point R equidistant from  [#permalink]

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Bunuel wrote:
In the xy-coordinate plane, is point R equidistant from points (-3,-3) and (1,-3) ?

Look at the diagram below: Notice, that the green line (x=-1) is the perpendicular bisector of the line segment with endpoints (-3,-3) and (1,-3), thus ANY point on this line will be equidistant from points (-3,-3) and (1,-3).

(1) The x-coordinate of point R is -1 --> point R is on the green line. Sufficient.
(2) Point R lies on the line y = -3 --> point R may or may not be on the green line. Not sufficient.

Attachment:
Equidistant points.png

Here in Statement(2),Point R lies on the line y = -3 then the only point on this line y=-3 equidistant from points is (-1,-3),.....so why not D?

From (2) point R can be ANY point on the line y = -3. If it's (-1, -3), then yes, R would be equidistant from points (-3,-3) and (1,-3) but if x-coordinate of point R is anything but -1, then R would NOT be equidistant from points (-3,-3) and (1,-3). For example, (-2, -3), (-1101, -3), ...
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Re: In the xy-coordinate plane, is point R equidistant from  [#permalink]

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Bunuel wrote:
In the xy-coordinate plane, is point R equidistant from points (-3,-3) and (1,-3) ?

Look at the diagram below: Notice, that the green line (x=-1) is the perpendicular bisector of the line segment with endpoints (-3,-3) and (1,-3), thus ANY point on this line will be equidistant from points (-3,-3) and (1,-3).

(1) The x-coordinate of point R is -1 --> point R is on the green line. Sufficient.
(2) Point R lies on the line y = -3 --> point R may or may not be on the green line. Not sufficient.

Attachment:
Equidistant points.png

Just feels bad even after drawing and everything I did not think it through and ended up with a wrong answer. Can't let this happen. Thanks for the explanation
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Re: In the xy-coordinate plane, is point R equidistant from  [#permalink]

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In the xy-coordinate plane, is point R equidistant from points (-3,-3) and (1,-3) ?

(1) The x-coordinate of point R is -1.
(2) Point R lies on the line y = -3.

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Re: In the xy-coordinate plane, is point R equidistant from  [#permalink]

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In the xy-coordinate plane, is point R equidistant from points (-3,-3) and (1,-3) ?

(1) The x-coordinate of point R is -1.
(2) Point R lies on the line y = -3.

Asked: In the xy-coordinate plane, is point R equidistant from points (-3,-3) and (1,-3) ?

Let the co-ordinates of R be (x,y)

(1) The x-coordinate of point R is -1.
x=-1
Distance of R(-1,y) from point (-3,-3) $$= \sqrt{2^2 + (y+3)^2}$$
Distance of R(-1,y) from point (1,-3) $$= \sqrt{2^2 + (y+3)^2}$$
Point R equidistant from points (-3,-3) and (1,-3)
SUFFICIENT

(2) Point R lies on the line y = -3.
y=-3
Distance of R(-1,y) from point (-3,-3) $$= \sqrt{(x+3)^2}=|x+3|$$
Distance of R(-1,y) from point (1,-3) $$= \sqrt{(x-1)^2}=|x-1|$$
Point R is NOT NECESSARILY equidistant from points (-3,-3) and (1,-3)
NOT SUFFICIENT

IMO A
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