Points A and B are at #(4 ,6 )# and #(8 ,3 )#, respectively. Point A is rotated counterclockwise about the origin by #pi # and dilated about point C by a factor of #1/2 #. If point A is now at point B, what are the coordinates of point C?

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Answer 1 · 2017-07-12T08:50:32

The coordinates of point #C# are #(20,12)#

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))((4),(6))=((-4),(-6))#

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

#vec(CB)=1/2 vec(CA')#

#((8-x),(3-y))=1/2((-4-x),(-6-y))#

So,

#8-x=1/2(-4-x)#

#16-2x=-4-x#

#x=20#

and

#3-y=1/2(-6-y)#

#6-2y=-6-y#

#y=12#

Therefore,

point #C=(20,12)#