Points A and B are at #(5 ,8 )# and #(7 ,1 )#, respectively. Point A is rotated counterclockwise about the origin by #pi # and dilated about point C by a factor of #5 #. If point A is now at point B, what are the coordinates of point C?

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Answer 1 · 2017-05-03T17:04:50

The point #C# is #=(-8,-41/4)#

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

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

Therefore, the trasformation of point #A# is

#A'=((-1,0),(0,-1))((5),(8))=((-5),(-8))#

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

#vec(CB)=5 vec(CA')#

#((7-x),(1-y))=5((-5-x),(-8-y))#

So,

#7-x=5(-5-x)#

#7-x=-25-5x#

#4x=-25-7=-32#

#x=-8#

and

#1-y=5(-8-y)#

#1-y=-40-5y#

#4y=-41#

#y=-41/4#

Therefore,

point #C=(-8,-41/4)#