Points A and B are at #(9 ,3 )# and #(4 ,8 )#, respectively. Point A is rotated counterclockwise about the origin by #pi # and dilated about point C by a factor of #3 #. If point A is now at point B, what are the coordinates of point C?

1 Answer
Jul 6, 2017

The coordinates of point #C# are #=(-31/2,-17/2)#

Explanation:

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),(3))=((-9),(-3))#

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

#vec(CB)=3vec(CA')#

#((4-x),(8-y))=3((-9-x),(-3-y))#

So,

#4-x=3(-9-x)#

#4-x=-27-3x#

#2x=-31#

#x=-31/2#

and

#8-y=3(-3-y)#

#8-y=-9-3y#

#2y=-17#

#y=-17/2#

Therefore,

The point #C=(-31/2,-17/2)#