Points A and B are at #(4 ,5 )# and #(7 ,2 )#, respectively. Point A is rotated counterclockwise about the origin by #(3pi)/2 # and dilated about point C by a factor of #1/2 #. If point A is now at point B, what are the coordinates of point C?

1 Answer
Apr 3, 2016

#C(9,8)#

Explanation:

Given: #A(4,5) and B(7,2)# A is rotated by #3pi/2# and dilated by a factor of 1/2 about a point C. After rotation and dilation A's new location is on B, that is #A^(RD)(7,2)#.
Required: the coordinates of C such that #A^(RD)(7,2) = B(7,2)#
Solution Strategy: a) Rotate #A#, by #R(3/2pi)#
b) Using Dilation about a #C# construct and knowing #A^(RD)# solve C
a) Rotation of #A# by #3/2pi#
#A^R = R(3/2pi) A#
#R(3/2pi)=[(cos(3/2pi),-sin(3/2pi) ),(sin(3/2pi),cos(3/2pi))]= [(0,1),(-1,0)]#
#A^R=[(0,1),(-1,0)] [(4),(5)] =[(5),(-4)]#
b) In order to dilate about C(x,y) we need to do do the following:
i) Translation #A^R# by #C(x,y)#
ii) Dilate by 1/2,
iii) Undo the translation:
Putting i), ii) and iii) we can wrtite:
# A^(RD) =[(7),(2)] = [(1/2(5-x)+x), (1/2(-4-y)+y ) ]# solve for #x and y#
#7=5/2-x/2+x; x=9#
#2=-2-y/2+y; y=8#
#C(9,8)#