Where does the rotation take the point (4, 1) if a 90 degree rotation counterclockwise about the origin takes (2, 0) to (0, 2) and (0, 3) to (-3, 0).

90 degree  about the origin (0,0) takes (2,0) to
(0,2)

90 degree about the origin takes (0,3) = (-3,
0)

To find where does the rotation of (4,1) about the
origin goes to.

Let A bethe point (4,1). and O is the
origin (0 , 0).

Then the slope of OA  = 
4/1

Length of OA = sqrt(4^2+1) =
sqr17.

When  A is rotated by 90 degree , it moves to a
place B in 2nd quadrant B (x,y).

OB = OA =
sqrt17.

x^2+y^2 =
17...(1)

Then the slope of B =
y/x

Product of the slope of OB andOA = -1 as the angle AOB
= 90 degrees.

(y/x)*4 = -1

Or
y = -4x.........(2).

Put y = -4x in
(1):

x^2+(-4x)^2 = 17

17x^2 =
1

x ^2 =  1.

x =
sqrt1.

x = 1 or x= -1.

When x
=  1, y = -4x = -4*1 =  -4, gives the position of rotation of 90 degree clockwise
direction.

When x = -1 , y = -(-1) = 4.
Or

B(x, y) =  B ( -1 , 4) is the position of A(1,4) after
90 degree anti clockwise  rotation.