Points A and B are at #(5 ,9 )# and #(8 ,6 )#, respectively. Point A is rotated counterclockwise about the origin by #pi # and dilated about point C by a factor of #4 #. If point A is now at point B, what are the coordinates of point C?

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Answer 1 · 2017-07-05T08:29:40

The point #C=(-28/3, -14)#

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))((5),(9))=((-5),(-9))#

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

#vec(CB)=4vec(CA')#

#((8-x),(6-y))=4((-5-x),(-9-y))#

So,

#8-x=4(-5-x)#

#8-x=-20-4x#

#3x=-28#

#x=-28/3#

and

#6-y=4(-9-y)#

#6-y=-36-4y#

#3y=-42#

#y=-14#

Therefore,

The point #C=(-28/3,-14)#