Points A and B are at #(8 ,3 )# and #(5 ,7 )#, respectively. Point A is rotated counterclockwise about the origin by #pi/2 # 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?

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Answer 1 · 2017-06-23T11:41:42

The point #C=(-7,17/2)#

The matrix of a rotation counterclockwise by #1/2pi# about the origin is

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

Therefore, the transformation of point #A# is

#A'=((0,-1),(1,0))((8),(3))=((-3),(8))#

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

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

#((5-x),(7-y))=3((-3-x),(8-y))#

So,

#5-x=3(-3-x)#

#5-x=-9-3x#

#2x=-14#

#x=-7#

and

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

#7-y=24-3y#

#2y=24-7#

#y=17/2#

Therefore,

point #C=(-7,17/2)#