Points A and B are at #(9 ,9 )# and #(7 ,6 )#, respectively. Point A is rotated counterclockwise about the origin by #pi # 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 · 2017-06-23T17:27:08

The point #C=(-25,-24)#

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

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

Therefore, the transformation of point #A# is

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

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

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

#((7-x),(6-y))=2((-9-x),(-9-y))#

So,

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

#7-x=-18-2x#

#x=-25#

and

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

#6-y=-18-2y#

#y=-18-6#

#y=-24#

Therefore,

The point #C=(-25,-24)#