Parallelogram OPQR lies in the xy-plane, as shown in the figure above.

Question

https://gmatclub.com/forum/download/file.php?id=41112 Parallelogram OPQR lies in the xy-plane, as shown in the figure above. The coordinates of point P are (2, 4) and the coordinates of point Q are (8, 6). What are the coordinates of point R ? A. (3, 2) B. (3, 3) C. (4, 4) D. (5, 2) E. (6, 2) Quant.jpg

EgmatQuantExpert · Answer

Solution

Given:  
• Parallelogram OPQR lies in the xy-plane
• Coordinate of point P = (2, 4)
• Coordinate of point Q = (8, 6)
• Coordinate of point O (which is the origin) = (0, 0)

To find:  
• Coordinate of point R = (x, y)

Approach and Working:   
• We know the diagonals of a parallelogram bisect each other at the point of intersection
 o Therefore, the midpoint of the diagonal PR = midpoint of the diagonal OQ 
 Or,  (\frac{(2+x)}{2}, \frac{(4+y)}{2}) = (\frac{(0+8)}{2}, \frac{(0+6)}{2})  
 Now comparing, we get: 2 + x = 8, and 4 + y = 6 
 Or x = 6 and y = 2

Hence, the correct answer is option E.

Answer: E

RohitGoyal1123 · Answer

Since OPQR is a parallelogram, opposite sides are parallel to each other
First lets find the slope of PQ = m =  \frac{(4-6)}{(2-8)} = \frac{1}{3} 
So y = mx + a
or 3y = x + c
using (2,4) and (8,6) we can find C = 10
So eq of PQ = > 3y = x+10

Now slope of OR will be the same as of PQ, the change will be the constant term
now OR passes through 0,0
so c in case of OR = 0

So Eq of OR is 3y = x

Only point satisfying is 6,2

Option  E

Hero8888 · Answer

The simplest approach: see attachement

Answer (E)

k1.saurabh · Answer

OPQR is parallelogram. 
OP is parallel to QR.
Difference between x co-ordinates of P and O is (2-0)=2
Hence difference between x co-ordinates between Q and R must be 2 i.e. 8-x=2.. so x is equal to 6.
There is only one option with x co-ordinate as 6 i.e. (6,2).. So answer is E..

minatminat · Answer

This is my approach:

The gradient of PQ and OR is equal, so i used: 
6-4/8-2 = Yr - 0/Xr-0
2/6 = Yr /Xr
 2Xr  = 6Yr ( Xr = 3Yr)

Gradient of PO and QR is equal, so: 
4-0 /2-0 = 6-Yr / 8 - Xr 
2 = 6-Yr/8-Xr 
16 - 2Xr = 6-Yx 
10+Yr =  2Xr

put both equations together 
6Ry = 10+Ry
5Ry = 10
Ry= 2

Rx therefore =6

Answer is E

generis · Answer

Quant2.png Find the slope of OP Slope of OP = Slope of QR Use the slope numbers and vertex Q's coordinates to find R's coordinates Use the slope numbers: count down vertically from Q's y-coordinate to get R's y-coordinate count left horizontally from Q's x-coordinate to get R's x-coordinate Side OP = Side QR (parallelogram property) Slope of OP = Slope of QR (parallel lines have identical slopes) From coordinates of O and P Slope of OP: \frac{rise}{run} = \frac{y_2-y_1}{x_2-x_1} = \frac{4-0}{2-0} = \frac{4}{2} So slope of QR = \frac{4}{2} The slope means that from R to Q, the change in y = 4 ("up 4') And change in x = 2 ("over 2") Just count the slope "backwards" from Q to find R ("down 4, leftwards 2") Vertex R will have a vertical y distance of 4 from vertex Q Vertex R will have a horizontal x distance of 2 from vertex Q Q is at (x,y) = (8,6) 4 down from y=6: = (6-4)= 2 = y-coordinate 2 leftwards from x=8: (8-2) = 6 = x-coordinate Coordinates of vertex R: (6, 2) Answer E

push12345 · Answer

Other method also

Opposite sides of parallelogram are equal

So distance of pq here is =√36+4 =√40

Let coordinate of R be (x,y),O(0,0)
Distance of OR =√x^2+y^2

Only option E gives value √40

Give kudos if it helps

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houston1980 · Answer

The solution is very simple . Since the figure is a parallelogram, Q-P= R-0
(8,6)-(2,4) = (R)-(0,0)
(6,2) = R
R=(6,2)

Hence   option E = (6,2)    is the answer.

BrentGMATPrepNow · Answer

KEY CONCEPT: Since OPQR is a parallelogram, we know that  sides PQ and OR are parallel AND the same length

Let's take a closer look at side PQ
 https://i.imgur.com/jkNYyis.png 
Notice that, to get from point P to point Q we must move 2 units UP and move 6 units RIGHT.

Since sides PQ and OR are parallel AND the same length, the same must apply to points O and R
So, if we start from point O (at 0,0) and move 2 units UP and move 6 units RIGHT, we must get to point R
 https://i.imgur.com/OzpCx37.png

From here, it we can determine the coordinates of point R
 https://i.imgur.com/cwzpWWp.png

Answer: E

RELATED VIDEO 
 https://www.youtube.com/watch?v=uQ356PWGq98

Himanshu453 · Answer

A much easier way is to just calculate the slope of the line PQ.
that comes out to be 1/3.
the other side of the parallelogram is parallel to the same line and the line passes through (0,0)
thus the ratio of the coordinates will be in the same ratio 1:3
thus E

bumpbot · Answer

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Parallelogram OPQR lies in the xy-plane, as shown in the figure above.

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Parallelogram OPQR lies in the xy-plane, as shown in the figure above.

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