Points A and B are at #(6 ,7 )# and #(3 ,9 )#, respectively. Point A is rotated counterclockwise about the origin by #(3pi)/2 # 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?

1 Answer
P dilip_k
Oct 21, 2016

we know , if in two dimension the rotation of a point (x,y) about origin by an angle #theta# in anticlockwise direction transforms its coordinates into (x',y') then

#x'=xcostheta-ysintheta#

#y'=xsintheta+ycostheta#

Here #theta=(3pi)/2#
#costheta=cos((3pi)/2)=0 and sintheta=sin((3pi)/2)=-1#

So transformed coordinates of #A->(6,7)# will be

#A'->((6*0-7*(-1)),(6*(-1))+7*0))#

#=(7,-6)#

Similarly transformed coordinates of #B->(3,9)# will be

#B'->((3*0-9*(-1)),(3*(-1))+9*0))#

#=(9,-3)#
Let the coordinates of center of dilation C be (h,k).

So A' on 5 times dilation about C will be transformed into

#A'_"5xdilated"=(5(7-h)+h,5(-6-k)+k)#

By the given condition #A'_"5xdilated"=B#
So

#5(7-h)+h=3=>4h=32=>h=8#

Again

#5(-6-k)+k=9=>4k=-39=>k=-39/4#

Hence coordinates of #C->(8,-39/4)#

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