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


Answer: E
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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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is (6,2) the square root of 40 simplified?

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)

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)

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.

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).

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

There's nice explanation provided by Brent.

However to answer ur query,

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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In Parallelogram, Opposite sides are parallel and equal

\(PQ = OR\)

\(PQ = \sqrt{(6-4)^2 + (8-2)^2} = \sqrt{2^2 + 6^2}\)

Since, \(OR\) have point \(O\) at origin, \(R\) must have coordinates \((6, 2)\) or \((2, 6)\)

Hence, option E
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