Points A and B are at #(4 ,9 )# 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?

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Answer 1 · 2017-05-07T08:32:24

The point #C# is #(-5/3,8)#

The matrix of a rotation counterclockwise by #3/2pi# about the origin is

#((0,1),(-1,0))#

Therefore, the trasformation of point #A# is

#A'=((0,1),(-1,0))((4),(9))=((9),(-4))#

Let point #C# be #(x,y)#, then

#vec(CB)=1/2 vec(CA')#

#((7-x),(2-y))=1/2((9-x),(-4-y))#

So,

#7-x=1/2(9-x)#

#14-2x=9-5x#

#3x=9-14=-5#

#x=-5/3#

and

#2-y=1/2(-4-y)#

#4-2y=-4-y#

#y=8#

Therefore,

point #C=(-5/3,8)#