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

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

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23 Jun 2016, 12:15
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Question Stats:

79% (01:07) correct 21% (01:44) wrong based on 380 sessions

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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)

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Md. Abdur Rakib

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

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24 Jun 2016, 01:16
10
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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Karishma
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Contact: bansal.karishma@gmail.com

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Joined: 23 Apr 2015
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Re: In the rectangular coordinate system shown above, points O, P, and Q r [#permalink]

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23 Jun 2016, 15:05
1
2

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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Joined: 03 Jun 2016
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In the rectangular coordinate system shown above, points O, P, and Q r [#permalink]

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24 Jun 2016, 02:20
2
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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Joined: 01 Mar 2017
Posts: 14
Re: In the rectangular coordinate system shown above, points O, P, and Q r [#permalink]

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07 Apr 2017, 10:33
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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Joined: 16 Oct 2010
Posts: 8128
Location: Pune, India
Re: In the rectangular coordinate system shown above, points O, P, and Q r [#permalink]

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09 Apr 2017, 22:42
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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Karishma
Private Tutor for GMAT
Contact: bansal.karishma@gmail.com

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Joined: 02 Sep 2009
Posts: 47109
Re: In the rectangular coordinate system shown above, points O, P, and Q r [#permalink]

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09 Apr 2017, 23:48
1
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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Joined: 10 Sep 2014
Posts: 71
In the rectangular coordinate system shown above, points O, P, and Q r [#permalink]

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23 Mar 2018, 04:44
I followed the midpoint property of co-ordinate geometry and got the same result. Is there any problem ? Bunuel, VeritasPrepKarishma, chetan2u, JeffTargetTestPrep, GMATPrepNow.
In the rectangular coordinate system shown above, points O, P, and Q r   [#permalink] 23 Mar 2018, 04:44
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