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# Parallelogram OPQR lies in the xy-plane

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Parallelogram OPQR lies in the xy-plane [#permalink]  27 Dec 2015, 19:42
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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)

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[Reveal] Spoiler: OA

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Re: Parallelogram OPQR lies in the xy-plane [#permalink]  27 Dec 2015, 20:00
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Here we have a parallelogram OPQR. So PQ has the same length as OR. PQ is (2,4) and (8,6).
Length of PQ = $$\sqrt{(8-2)^2 + (6 -4)^2}$$ .

PQ =$$\sqrt{(6)^2 + (2)^2}$$=$$\sqrt{40}$$.

Only option E (6,2) holds correct.
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Re: Parallelogram OPQR lies in the xy-plane [#permalink]  29 Mar 2016, 09:26
is (6,2) the square root of 40 simplified?
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Re: Parallelogram OPQR lies in the xy-plane [#permalink]  27 Apr 2016, 08:27
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It is a parallelogram. If the distance between O to P in “X”axis is 2, the distance between R to Q in “X” axis must be 2. So, Distance between O to R in “X” axis will be 8-2= 6. Similarly, If the distance between O to P in “Y”axis is 4, the distance between Q to R in “Y” axis must be 4. So, Distance between O to R in “Y” axis will be 6-4= 2. The point at R= (6,2)
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Re: Parallelogram OPQR lies in the xy-plane [#permalink]  29 May 2016, 23:18
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As OPQR is a parallelogram, OQ and PR bisects each other. The mid point of OQ is (4,3). Now the slope of PR is -1/2. So the coordinate of R is (6,2)
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Re: Parallelogram OPQR lies in the xy-plane [#permalink]  15 Sep 2017, 00:14
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I have used the fact that PQ and OR must be parallel since the figure is a parallelogram, then the slope of PQ must be the same as OR. The slope of PQ is equal to 1/3, thus, given that O=(0,0), R must be equal to (6,2) in order to have a slope of 1/3, i.e. (2-0)/(6-0) = 1/3.
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Re: Parallelogram OPQR lies in the xy-plane [#permalink]  20 Mar 2018, 13:27
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sandy wrote:

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)

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

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

From here, it we can determine the coordinates of point R

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Re: Parallelogram OPQR lies in the xy-plane [#permalink]  27 Jun 2018, 20:47
The slope of the line from the origin to point P is the same as the slope of the line from point Q to R because we have a parallelgram. The same is true for the lines between points P and Q and the origin and point R. The slope of the line from the origin to P is 2, so it follows that the slope of the line between between Q and R is also 2. Now we can determine the equations of the lines between the origin and R and also between Q and R. Given points P and Q, it is easy to see that the slope line between these points is 1/3. It follows that the slope of the line between the origin and R is also 1/3. Hence, the equation of the line from the origin to R is y=1/3X. Using the fact that the slope of the line between Q and R is 2 and we are given that the coordinates of Q are (8,6), we can determine the equation of the line between Q and R to be Y=2X-10 using the point-slope method. We can determine the coordinates of R by setting the equations of the lines Y=1/3X and Y=2X-10 together. Solve for X: 1/3X=2X-10, So X is 6. Plug 6 into Y=1/3X or Y=2X-10 to get Y=2. Answer: (6,2).
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Re: Parallelogram OPQR lies in the xy-plane [#permalink]  17 Feb 2019, 06:26
sandy wrote:
Here we have a parallelogram OPQR. So PQ has the same length as OR. PQ is (2,4) and (8,6).
Length of PQ = $$\sqrt{(8-2)^2 + (6 -4)^2}$$ .

PQ =$$\sqrt{(6)^2 + (2)^2}$$=$$\sqrt{40}$$.

Only option E (6,2) holds correct.

Hello ıf anyone can clarify how square root of 40 , has been translated to E.
Thanks
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Re: Parallelogram OPQR lies in the xy-plane [#permalink]  17 Feb 2019, 07:17
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SKH121 wrote:
sandy wrote:
Here we have a parallelogram OPQR. So PQ has the same length as OR. PQ is (2,4) and (8,6).
Length of PQ = $$\sqrt{(8-2)^2 + (6 -4)^2}$$ .

PQ =$$\sqrt{(6)^2 + (2)^2}$$=$$\sqrt{40}$$.

Only option E (6,2) holds correct.

Hello ıf anyone can clarify how square root of 40 , has been translated to E.
Thanks

There's nice explanation provided by Brent.

I hope ur aware in parallelogram opposite sides are equal,

as the side PQ = $$\sqrt{40}$$= side OR

Now we the need the co ordinates of R such that the distance OR = $$\sqrt{40}$$,

option E is (6, 2)

meaning the distance from the origin to the Point R = $$\sqrt{(0 - 6)^2 + (0 - 2)^2} = \sqrt{40}$$
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Re: Parallelogram OPQR lies in the xy-plane   [#permalink] 17 Feb 2019, 07:17
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