Point A is at #(4 ,-2 )# and point B is at #(2 ,-3 )#. Point A is rotated #pi/2 # clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?

Finding the new coordinate
With the #x# -coordinate positive and #y# -coordinate negative the point #A# used to lie in the fourth quadrant of the Cartesian plane (the bottom-right corner); rotating it clockwise by #pi/2# (or equivalently by #90^"o"# ) would move it to the third quadrant (the bottom-left corner) such that both its #x# and #y# -coordinates are now positive.

Rotating a point with respect to the origin by an odd integer multiple of #pi/2# or #90^"o"# would swap its #x# and #y# -coordinates. So now point #A# would have coordinates #(-2, -4)#.

Finding and comparing the distance between #A# and #B#
Now that we have got the coordinates of #A# before and after the rotation, so we can apply the Pythagorean theorem in the Cartesian to find the distance between point #A# and #B#:

Before the rotation:
#A_i B = sqrt((x_(A_i)- x_B)^2+(y_(A_i)- y_B)^2)=sqrt((4-2)^2+(-2-(-3))^2)=sqrt(5)#

After the rotation:
#A_fB = sqrt((x_(A_f)- x_B)^2+(y_(A_f)- y_B)^2)=sqrt((-2-2)^2+(-4-(-3))^2)=1#

Therefore the difference would be
#abs(A_i B-A_f B)=sqrt(5)-1#

References/ See also:
"Distance Between Two Points", Massey University,
http://mathsfirst.massey.ac.nz/Algebra/PythagorasTheorem/pythapp.htm
Wikipedia contributors. "Quadrant (plane geometry)." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 22 Feb. 2018. https://en.wikipedia.org/wiki/Quadrant_(plane_geometry) 14 Mar. 2018.