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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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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)
[Reveal] Spoiler: OA

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

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24 Jun 2016, 01:16
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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).
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 03 Jun 2016 Posts: 10 In the rectangular coordinate system shown above, points O, P, and Q r [#permalink] ### Show Tags 24 Jun 2016, 02:20 2 This post received KUDOS 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). Intern Joined: 01 Mar 2017 Posts: 14 Re: In the rectangular coordinate system shown above, points O, P, and Q r [#permalink] ### Show Tags 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. Answer (E) 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). Answer (E) 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? Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8024 Location: Pune, India Re: In the rectangular coordinate system shown above, points O, P, and Q r [#permalink] ### Show Tags 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. Answer (E) 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). Answer (E) 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. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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

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09 Apr 2017, 23:48
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Expert's post
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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23 Mar 2018, 04:44
I followed the midpoint property of co-ordinate geometry and got the same result. Is there any problem ? Bunuel
Re: In the rectangular coordinate system shown above, points O, P, and Q r   [#permalink] 23 Mar 2018, 04:44
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