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

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Answer 1 · 2017-06-24T06:26:14

The coordinates of point #C# are #(1,-6)#

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

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

Therefore, the transformation of point #A# is

#A'=((0,1),(-1,0))((2),(4))=((4),(-2))#

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

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

#((7-x),(2-y))=2((4-x),(-2-y))#

So,

#7-x=2(4-x)#

#7-x=8-2x#

#x=1#

and

#2-y=2(-2-y)#

#2-y=-4-2y#

#y=-4-2#

#y=-6#

Therefore,

point #C=(1,-6)#