Points A and B are at #(3 ,5 )# and #(6 ,1 )#, 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 · 2017-06-08T08:59:48

The coordinates of point #C=(9/2,-5)#

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))((3),(5))=((5),(-3))#

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

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

#((6-x),(1-y))=3((5-x),(-3-y))#

So,

#6-x=3(5-x)#

#15-3x=6-x#

#2x=9#

#x=9/2#

and

#1-y=3(-3-y)#

#-9-3y=1-y#

#2y=-10#

#y=-5#

Therefore,

point #C=(9/2,-5)#