The two given coordinates (√3, 1) should remind us of the special 30-60-90 right triangle.
If we draw a line from the point that is perpendicular to the x-axis, we get a right triangle.
This means we can apply the Pythagorean theorem to determine that the length of the line segment is 2.
At this point, we can see the special 30-60-90 right triangle hiding in the diagram.



When we rotate the line segment 90 degrees, the length of the line segment is still 2.



If we draw a line from the new point to the x-axis, we get another a right triangle.
More importantly, we can see that our new right triangle is also a 30-60-90 right triangle, which means it has the following lengths.


From here we can see that (-1, √3) are the coordinates of the new point.
The x-coordinate of the new point is -1, which means Quantities A and B are equal.

Answer: C

Cheers,
Brent
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since rotation is 90deg counterclockwise
the coordinates of point P(x,y) will become (-y,x) for Q on the new line OQ (rotated line)
i.e if coordinates of P are (x,y) then after 90deg rotation of OP, on line OQ, the coordinates of point Q will be (-y,x)
in this case coordinates of P are (root3,1)
hence Q will have coordinates (-1,root3)
now x coordinate of Q is -1 which is equal to Quantity B
hence the answer is C

Ok I kinda remember the rules of mirroring from physics but In application I cannot understand why x and y will be switched.

since just its -ve of x and nothing else is changed

Since it is rotated counterclockwise (opposite direction to clockwise direction), the coordinates of OQ will be (-x,y) and not (-y,x). The current value of x is sqt3, which is equal to 1.732. Thus, new coordinates of x will be -1.732, which is less than -1. Thus, the answer should be B.
If the question asked about the y-coordinate, then the answer will be C.

Please clarify, I have the same doubt

@carcass?

Great explanation Sir
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Since OP is rotated counterclockwise through an angle 90 degree, which is OQ. Hence OP & OQ are perpendicular to each other.As a result the slope of line OP= 1/√3 and line OQ=-√3/1. Therefore X coordinate of point Q is -1.

Answer C

Sir, a great explanation in the first place! But, still I'm not able to understand completely. When rotated 90 degrees, how did the corresponding angles 30 and 60 interchange? Can you explain the thought process behind this?

First, I used the fact that angles on a line add up to 180°.
We already had two angles (90° and 30°), which means the remaining angle must be 60°.

Does that help?
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Might be useful for easy rotations:

90° clockwise rotation: (x,y) becomes (y,-x)

90° counterclockwise rotation: (x,y) becomes (-y,x)

180° clockwise and counterclockwise rotation: (x, y) becomes (-x,-y)

270° clockwise rotation: (x,y) becomes (-y,x)

270° counterclockwise rotation: (x,y) becomes (y,-x)

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