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In the rectangular coordinate system shown above, points O, P, and Q r  [#permalink]

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Question Stats: 80% (01:36) correct 20% (01:59) wrong based on 1019 sessions

### HideShow timer Statistics In the rectangular coordinate system shown above, points O, P, and Q represent the sites of three proposed housing developments. If a fire station can be built at any point in the coordinate system, at which point would it be equidistant from all three developments?

A. (3,1)
B. (1,3)
C. (3,2)
D. (2,2)
E. (2,3)

OG Q 2017(Book Question: 24)

Attachment: 24xj68n.jpg [ 11.54 KiB | Viewed 8837 times ]
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In the rectangular coordinate system shown above, points O, P, and Q r  [#permalink]

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7
AbdurRakib wrote: In the rectangular coordinate system shown above, points O, P, and Q represent the sites of three proposed housing developments. If a fire station can be built at any point in the coordinate system, at which point would it be equidistant from all three developments?
A. (3,1)
B. (1,3)
C. (3,2)
D. (2,2)
E. (2,3)

OG Q 2017(Book Question: 24)

All points equidistant from O and Q lie on the line x = 2 so the fire station should lie on this line.
All points equidistant from O and P lie on the line y = 3 so the fire station should lie on this line too.
These two intersect at (2, 3) and that will be the point equidistant from all 3 points.

Or

You can think of the question in terms of the perpendicular bisectors of triangle OPQ. Their point of intersection will be equidistant from all three vertices.
Again the perpendicular bisector of OQ will be x = 2 and of OP will be y = 3. They will intersect at (2, 3).
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Re: In the rectangular coordinate system shown above, points O, P, and Q r  [#permalink]

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5
5

the point we're asked is the Centroid of the triangle (scalene). Let it be (x,y). Then it should have same distance from all three points.So using the Pythagoras theorem of distance between 2 points:
distance from origin = distance from (4,0)
$$\sqrt{(x^2 + y^2)} = \sqrt{((4-x)^2 + y^2)}$$

distance from origin = distance from (0,6)
$$\sqrt{(x^2 + y^2)} = \sqrt{(x^2 + (6-y)^2)}$$

Solve these 2 for x and y and it will give (2,3)

(Share a kudo's , if you like the explanation)
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Re: In the rectangular coordinate system shown above, points O, P, and Q r  [#permalink]

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The point should be the circumcentre of the triangle and in right angled triangle circumcentre lies on hypotenuse and is mid-point of it . Therefore it can be directly seen that mid point of the hypotenuse will be E.(2,3).
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In the rectangular coordinate system shown above, points O, P, and Q r  [#permalink]

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VeritasPrepKarishma wrote:
AbdurRakib wrote: In the rectangular coordinate system shown above, points O, P, and Q represent the sites of three proposed housing developments. If a fire station can be built at any point in the coordinate system, at which point would it be equidistant from all three developments?
A. (3,1)
B. (1,3)
C. (3,2)
D. (2,2)
E. (2,3)

OG Q 2017(Book Question: 24)

All points equidistant from O and Q lie on the line x = 2 so the fire station should lie on this line.
All points equidistant from O and P lie on the line y = 3 so the fire station should lie on this line too.
These two intersect at (2, 3) and that will be the point equidistant from all 3 points.

Or

You can think of the question in terms of the perpendicular bisectors of triangle OPQ. Their point of intersection will be equidistant from all three vertices.
Again the perpendicular bisector of OQ will be x = 2 and of OP will be y = 3. They will intersect at (2, 3).

Is it a general fact, that bisectors of the two legs of a right triangle will cut the hypotenuse exactly in half, in other words, intersect the hypotenuse at its midpoint?
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In the rectangular coordinate system shown above, points O, P, and Q r  [#permalink]

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melin94 wrote:
VeritasPrepKarishma wrote:
AbdurRakib wrote: In the rectangular coordinate system shown above, points O, P, and Q represent the sites of three proposed housing developments. If a fire station can be built at any point in the coordinate system, at which point would it be equidistant from all three developments?
A. (3,1)
B. (1,3)
C. (3,2)
D. (2,2)
E. (2,3)

OG Q 2017(Book Question: 24)

All points equidistant from O and Q lie on the line x = 2 so the fire station should lie on this line.
All points equidistant from O and P lie on the line y = 3 so the fire station should lie on this line too.
These two intersect at (2, 3) and that will be the point equidistant from all 3 points.

Or

You can think of the question in terms of the perpendicular bisectors of triangle OPQ. Their point of intersection will be equidistant from all three vertices.
Again the perpendicular bisector of OQ will be x = 2 and of OP will be y = 3. They will intersect at (2, 3).

Is it a general fact, that bisectors of the two legs of a right triangle will cut the hypotenuse exactly in half, in other words, intersect the hypotenuse at its midpoint?

Yes, that is correct. The circumcenter of a right triangle (the point of intersection of perpendicular bisectors) will bisect the hypotenuse.
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Re: In the rectangular coordinate system shown above, points O, P, and Q r  [#permalink]

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VeritasPrepKarishma wrote:
melin94 wrote:
VeritasPrepKarishma wrote:

All points equidistant from O and Q lie on the line x = 2 so the fire station should lie on this line.
All points equidistant from O and P lie on the line y = 3 so the fire station should lie on this line too.
These two intersect at (2, 3) and that will be the point equidistant from all 3 points.

Or

You can think of the question in terms of the perpendicular bisectors of triangle OPQ. Their point of intersection will be equidistant from all three vertices.
Again the perpendicular bisector of OQ will be x = 2 and of OP will be y = 3. They will intersect at (2, 3).

Is it a general fact, that bisectors of the two legs of a right triangle will cut the hypotenuse exactly in half, in other words, intersect the hypotenuse at its midpoint?

Yes, that is correct. The circumcenter of a right triangle (the point of intersection of perpendicular bisectors) will bisect the hypotenuse.

Another way to phrase this property is: in a right triangle, the median drawn to the hypotenuse has the measure half the hypotenuse.
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Re: In the rectangular coordinate system shown above, points O, P, and Q r  [#permalink]

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I followed the midpoint property of co-ordinate geometry and got the same result. Is there any problem ? Bunuel, VeritasPrepKarishma, chetan2u, JeffTargetTestPrep, GMATPrepNow.
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GMAT 1: 640 Q47 V32 GMAT 2: 710 Q49 V38 Re: In the rectangular coordinate system shown above, points O, P, and Q r  [#permalink]

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All three points would be on the periphery of a circle while the fire station would be at the center.
So, (X-a)^2+(Y-b)^2=C^2, where (a,b) are the three points.
Solving three equations we get (X,Y) as (2,3).
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Re: In the rectangular coordinate system shown above, points O, P, and Q r  [#permalink]

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Use Centroid of Triangle formula.

Centroid of Triangle is (x1+x2+x3)/3, (y1+y2+y3)/3

here, X and Y are points of triangle.
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Re: In the rectangular coordinate system shown above, points O, P, and Q r  [#permalink]

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sunngupt11 wrote:
Use Centroid of Triangle formula.

Centroid of Triangle is (x1+x2+x3)/3, (y1+y2+y3)/3

here, X and Y are points of triangle.

Well sunngupt11,

If you use centroid formula you get$$\frac {4}{3}$$, $$\frac {6}{3}$$

Also centroid of triangle lies inside the triangle. If you look at the responses above many have solved this using circumradius or perpendicular bisectors of triangle.

Now the three perpendicular bisectors of triangle can meet inside the triangle or outside the triangle .

I guess what must have confused you is that centroid divides the media in a ratio of 2:1. but that dosen't mean all the medians of a triangle will be of equal length.
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Re: In the rectangular coordinate system shown above, points O, P, and Q r  [#permalink]

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1
knowing that it is a right triangle, is using the midpoint formula suitable?

(X1 + x2)/ 2, (y1+y2)/2

(0 + 4) / 2 , (0 + 6) / 2

2,3
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In the rectangular coordinate system shown above, points O, P, and Q r  [#permalink]

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AbdurRakib wrote: In the rectangular coordinate system shown above, points O, P, and Q represent the sites of three proposed housing developments. If a fire station can be built at any point in the coordinate system, at which point would it be equidistant from all three developments?
A. (3,1)
B. (1,3)
C. (3,2)
D. (2,2)
E. (2,3)

OG Q 2017(Book Question: 24)

All points equidistant from O and Q lie on the line x = 2 so the fire station should lie on this line.
All points equidistant from O and P lie on the line y = 3 so the fire station should lie on this line too.
These two intersect at (2, 3) and that will be the point equidistant from all 3 points.

Or

You can think of the question in terms of the perpendicular bisectors of triangle OPQ. Their point of intersection will be equidistant from all three vertices.
Again the perpendicular bisector of OQ will be x = 2 and of OP will be y = 3. They will intersect at (2, 3).

Responding to a pm:
Quote:
Let the point be A(x,y)

Considering that it is equidistant from O, P, Q,

Distance(OA) = Distance(PA) = Distance (QA)

x^2+y^2= x^2+ (y-6)^2=(x-4)^2+y^2

Solving the above, gives

(y-6)^2= (x-4)^2

Option B (3,1) satisfies the above.

Not sure how you get (y-6)^2= (x-4)^2

On solving, I get y^2 = (y - 6)^2 which gives y = 3
and x^2 = (x - 4)^2 which gives x = 2
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Re: In the rectangular coordinate system shown above, points O, P, and Q r  [#permalink]

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AbdurRakib wrote: In the rectangular coordinate system shown above, points O, P, and Q represent the sites of three proposed housing developments. If a fire station can be built at any point in the coordinate system, at which point would it be equidistant from all three developments?
A. (3,1)
B. (1,3)
C. (3,2)
D. (2,2)
E. (2,3)

OG Q 2017(Book Question: 24)

Attachment:
24xj68n.jpg

Point will be equidistant from all three developments if it lies on the perpendicular bisector of line joining PQ i.e, MID POINT OF PQ(2,3) as per the given options.
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In the rectangular coordinate system shown above, points O, P, and Q r  [#permalink]

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AbdurRakib wrote: In the rectangular coordinate system shown above, points O, P, and Q represent the sites of three proposed housing developments. If a fire station can be built at any point in the coordinate system, at which point would it be equidistant from all three developments?
A. (3,1)
B. (1,3)
C. (3,2)
D. (2,2)
E. (2,3)

OG Q 2017(Book Question: 24)

All points equidistant from O and Q lie on the line x = 2 so the fire station should lie on this line.
All points equidistant from O and P lie on the line y = 3 so the fire station should lie on this line too.
These two intersect at (2, 3) and that will be the point equidistant from all 3 points.

Or

You can think of the question in terms of the perpendicular bisectors of triangle OPQ. Their point of intersection will be equidistant from all three vertices.
Again the perpendicular bisector of OQ will be x = 2 and of OP will be y = 3. They will intersect at (2, 3).

Can we find mid point?

(0+4)/2 and (6+0)/2
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Re: In the rectangular coordinate system shown above, points O, P, and Q r  [#permalink]

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Can we approach the problem like below?

Since it is asked us to find a point in the coordinate system, which must be equal distances to the other points, can't we use the distance between two points or difference between 2 points? In differences, the only right answer will be E. Re: In the rectangular coordinate system shown above, points O, P, and Q r   [#permalink] 21 Apr 2020, 02:13

# In the rectangular coordinate system shown above, points O, P, and Q r  