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# Points A = (3,9), B = (1,1), C = (5,3), and D=(a,b) lie in the first

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Joined: 02 Sep 2009
Posts: 59725
Points A = (3,9), B = (1,1), C = (5,3), and D=(a,b) lie in the first  [#permalink]

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18 Mar 2019, 02:58
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25% (01:25) correct 75% (03:19) wrong based on 8 sessions

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Points A = (3,9), B = (1,1), C = (5,3), and D=(a,b) lie in the first quadrant and are the vertices of quadrilateral ABCD. The quadrilateral formed by joining the midpoints of AB, BC, CD, and DA is a square. What is the sum of the coordinates of point D?

(A) 7
(B) 9
(C) 10
(D) 12
(E) 16

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Re: Points A = (3,9), B = (1,1), C = (5,3), and D=(a,b) lie in the first  [#permalink]

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18 Mar 2019, 04:25
Bunuel wrote:
Points A = (3,9), B = (1,1), C = (5,3), and D=(a,b) lie in the first quadrant and are the vertices of quadrilateral ABCD. The quadrilateral formed by joining the midpoints of AB, BC, CD, and DA is a square. What is the sum of the coordinates of point D?

(A) 7
(B) 9
(C) 10
(D) 12
(E) 16

Check the video solution as mentioned below

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Screenshot 2019-03-18 at 4.51.57 PM.png [ 864.39 KiB | Viewed 548 times ]

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Re: Points A = (3,9), B = (1,1), C = (5,3), and D=(a,b) lie in the first  [#permalink]

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18 Mar 2019, 06:11
1
Bunuel wrote:
Points A = (3,9), B = (1,1), C = (5,3), and D=(a,b) lie in the first quadrant and are the vertices of quadrilateral ABCD. The quadrilateral formed by joining the midpoints of AB, BC, CD, and DA is a square. What is the sum of the coordinates of point D?

(A) 7
(B) 9
(C) 10
(D) 12
(E) 16

It can get done in a minute if you follow the logic discussed here:
https://www.veritasprep.com/blog/2013/0 ... ry-part-i/
https://www.veritasprep.com/blog/2013/0 ... y-part-ii/

Now, look at the diagram below:

Attachment:

Screenshot 2019-03-18 at 18.40.37.png [ 503.47 KiB | Viewed 510 times ]

Plot the 3 co-ordinates (A, B and C) of the given quad and find their mid points to get two co-ordinates of the square (N and M). Now MP will be another side of the square such that it is perpendicular and of the same length as MN.

Hence MX = 3 = MY
NX = 1 = PY

So point P will be 3 to the right of M and 1 above M i.e. at (5, 6).
P will be the mid point of AD and hence D will be (7, 3).

Sum of the coordinates of D = 7+3 = 10

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Karishma
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Joined: 10 Dec 2016
Posts: 21
Re: Points A = (3,9), B = (1,1), C = (5,3), and D=(a,b) lie in the first  [#permalink]

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19 Mar 2019, 02:59
Bunuel wrote:
Points A = (3,9), B = (1,1), C = (5,3), and D=(a,b) lie in the first quadrant and are the vertices of quadrilateral ABCD. The quadrilateral formed by joining the midpoints of AB, BC, CD, and DA is a square. What is the sum of the coordinates of point D?

(A) 7
(B) 9
(C) 10
(D) 12
(E) 16

It can get done in a minute if you follow the logic discussed here:
https://www.veritasprep.com/blog/2013/0 ... ry-part-i/
https://www.veritasprep.com/blog/2013/0 ... y-part-ii/

Now, look at the diagram below:

Attachment:
Screenshot 2019-03-18 at 18.40.37.png

Plot the 3 co-ordinates (A, B and C) of the given quad and find their mid points to get two co-ordinates of the square (N and M). Now MP will be another side of the square such that it is perpendicular and of the same length as MN.

Hence MX = 3 = MY
NX = 1 = PY

So point P will be 3 to the right of M and 1 above M i.e. at (5, 6).
P will be the mid point of AD and hence D will be (7, 3).

Sum of the coordinates of D = 7+3 = 10

Hi

I am unable to understand the below points.

Hence MX = 3 = MY
NX = 1 = PY

So point P will be 3 to the right of M and 1 above M i.e. at (5, 6).

Thanks
Sekhar
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9876
Location: Pune, India
Re: Points A = (3,9), B = (1,1), C = (5,3), and D=(a,b) lie in the first  [#permalink]

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19 Mar 2019, 03:12
sekharm2389 wrote:
Bunuel wrote:
Points A = (3,9), B = (1,1), C = (5,3), and D=(a,b) lie in the first quadrant and are the vertices of quadrilateral ABCD. The quadrilateral formed by joining the midpoints of AB, BC, CD, and DA is a square. What is the sum of the coordinates of point D?

(A) 7
(B) 9
(C) 10
(D) 12
(E) 16

It can get done in a minute if you follow the logic discussed here:
https://www.veritasprep.com/blog/2013/0 ... ry-part-i/
https://www.veritasprep.com/blog/2013/0 ... y-part-ii/

Now, look at the diagram below:

Attachment:
Screenshot 2019-03-18 at 18.40.37.png

Plot the 3 co-ordinates (A, B and C) of the given quad and find their mid points to get two co-ordinates of the square (N and M). Now MP will be another side of the square such that it is perpendicular and of the same length as MN.

Hence MX = 3 = MY
NX = 1 = PY

So point P will be 3 to the right of M and 1 above M i.e. at (5, 6).
P will be the mid point of AD and hence D will be (7, 3).

Sum of the coordinates of D = 7+3 = 10

Hi

I am unable to understand the below points.

Hence MX = 3 = MY
NX = 1 = PY

So point P will be 3 to the right of M and 1 above M i.e. at (5, 6).

Thanks
Sekhar

I am guessing you did not go through the two links I gave above. The logic of why MY will be 3 and PY will be 1 is explained in them in detail. Without checking the two posts, you will not be able to understand this solution. Please do go through them and then let me know if it makes sense.
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Veritas Prep GMAT Instructor

Re: Points A = (3,9), B = (1,1), C = (5,3), and D=(a,b) lie in the first   [#permalink] 19 Mar 2019, 03:12
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