Points A and B are at #(3 ,9 )# and #(9 ,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 · 2017-03-28T07:12:49

The coordinates of #C=(-3,-20)#

The point #A=(3,9)# becomes (after a rotation of #3/2pi#)

#A'=(3,-9)#

Let the center of the dilation be #C=(x,y)#

#B=(9,2)#

We perform this with vectors

#vecCB=2vec(CA')#

#<9-x,2-y> = 2<3-x,-9-y>#

Therefore,

#9-x=2(3-x)=6-2x#

#2x-x=6-9#

#x=-3#

and

#2-y=2(-9-y)=-18-2y#

#2y-y=-18-2#

#y=-20#

So,

#C=(-3,-20)#