Points A and B are at #(9 ,2 )# and #(2 ,5 )#, respectively. Point A is rotated counterclockwise about the origin by #(3pi)/2 # and dilated about point C by a factor of #3 #. If point A is now at point B, what are the coordinates of point C?

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Answer 1 · 2017-06-02T09:46:39

The coordinates of the point #C=(2,-16)#

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))((9),(2))=((2),(-9))#

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

#vec(CB)=3 vec(CA')#

#((2-x),(5-y))=3((2-x),(-9-y))#

So,

#2-x=3(2-x)#

#6-3x=2-x#

#2x=4#

#x=2#

and

#5-y=3(-9-y)#

#-27-3y=5-y#

#2y=-32#

#y=-16#

Therefore,

point #C=(2,-16)#