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Re: The point (-3, 2) is rotated 90° clockwise around the origin to point [#permalink]
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Bunuel wrote:
The point (-3, 2) is rotated 90° clockwise around the origin to point B. Point B is then reflected over the line x = y to point C. What are the coordinates of C?

(A) (-3, -2)
(B) (-2, -3)
(C) (2, -3)
(D) (2, 3)
(E) (3, 2)


Given: The point (-3, 2) is rotated 90° clockwise around the origin to point B. Point B is then reflected over the line x = y to point C.
Asked: What are the coordinates of C?

Attachment:
Screenshot 2020-07-14 at 10.49.02 AM.png
Screenshot 2020-07-14 at 10.49.02 AM.png [ 26.13 KiB | Viewed 21447 times ]


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The point (-3, 2) is rotated 90° clockwise around the origin to point [#permalink]
Reflection 1:

The way I understood 90 degree rotations was by the “tipping the rectangle” method.

It sounds goofy, but:


Start with Original Point (-3, 2)

Since we are rotating 90 degrees about the Origin (0 , 0) ——> make a rectangle with these 2 points

Label (-3 , 2) as point A and Origin as Point O

If we join the horizontal and vertical lines to the Y and X axis, respectively, we end up with a rectangle that is 3 units along the Negative X Axis and 2 units along the Positive Y Axis


Now imagine lifting the rectangle up and pushing it from quadrant 2 into quadrant 1 - i.e, a 90 degree rotation clockwise about the Origin


This would give us a rectangle of 2 units along the positive X axis and 3 units along positive Y axis. Point A would now move to the upper right corner of this new rectangle that we pushed over ———> this will be point (2 , 3)

You can confirm visually by:

-connecting original point A at (-3 ,2) with the Origin

-and connecting new image point A at (2 , 3)

these 2 points and the origin will create a 90 degree angle about the Origin (both lines are diagonals of their respective rectangles)


Or, the explanation is too long winded, you can follow the Rule:

For 90 degree rotations clockwise about the Origin: (X , Y) ———>becomes image point of (Y , -X)

(-3 ,2) becomes (2, 3)

So the first rotation gives us point (2 , 3)


Reflection 2:

Line Y = X is the Line that passes through the origin and creates a 45 degree angle with the X axis

The original point of (2 , 3) and the Image Point will always be equidistant from the Mirror Line over which the original point is reflected.

To visualize it in this problem:

(2 ,2) and (3 , 3) are both on line Y = X

Point (2 , 3) will be +1 unit above (2 ,2) and + 1 unit to the left of (3 , 3)

This creates a right triangle with sides of 1 and 1

To find the image point reflected over Y = X, we make the opposite moves from these points (reversed) on Line Y = X:

from (2 , 2) we move 1 unit to the right: X coordinate = 3

From (3 , 3) we move 1 unit down: Y coordinate = 2


Final Point is (3 , 2)


Or the rule:

Reflecting point (X , Y) over the line given by Y = X ————> image point will be (Y , X)

In other words, just rearrange the coordinates



Answer E

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Re: The point (-3, 2) is rotated 90° clockwise around the origin to point [#permalink]
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Bunuel wrote:
The point (-3, 2) is rotated 90° clockwise around the origin to point B. Point B is then reflected over the line x = y to point C. What are the coordinates of C?

(A) (-3, -2)
(B) (-2, -3)
(C) (2, -3)
(D) (2, 3)
(E) (3, 2)



We have discussed this concept in detail here:

https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2013/0 ... ry-part-i/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2013/0 ... y-part-ii/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2013/0 ... -part-iii/

When the point (-3, 2) is rotated clockwise 90 degrees, the point lies in first quadrant and co-ordinates of x and y flip. So you get the point (2, 3).

When you reflect (2, 3) over the line x = y, this is like having a square with one vertex at (2, 3), one at (2, 2), one at (3, 3) so the fourth vertex will be at (3, 2).

Answer (E)
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Re: The point (-3, 2) is rotated 90° clockwise around the origin to point [#permalink]
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Re: The point (-3, 2) is rotated 90° clockwise around the origin to point [#permalink]
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