Points A and B are at #(4 ,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 #4 #. If point A is now at point B, what are the coordinates of point C?

Question

Points A and B are at #(4   ,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 #4  #.  If point A is now at point B, what are the coordinates of point C?

Claire Akin · Accepted Answer

#color(indigo)("Coordinates of "C((x),(y)) = ((19),(-43))#

#A(4,7), B(3,9), "counterclockwise rotation " #(3pi)/2#, "dilation factor" 4#

New coordinates of A after #(3pi)/2# counterclockwise rotation

#A(4,7) rarr A' (7,-4)#

#vec (BC) = (4) vec(A'C)#

#b - c = (4)a' - (4)c#

#c = (4)a' - (3)b#

#color(indigo)(C((x),(y)) = (4)((7),(-4)) - (3)((3),(9)) = ((19),(-43))#

Mason Adams · Answer

#C=(25/3,-25/3)# #"under a counterclockwise rotation about the origin of "(3pi)/2# #• " a point "(x,y)to(y,-x)# #A(4,7)toA'(7,-4)" where A' is the image of A"# #vec(CB)=color(red)(4)vec(CA')# #ulb-ulc=4(ula'-ulc)# #ulb-ulc=4ula'-4ulc# #3ulc=4ula'-ulb# #color(white)(3ulc)=4((7),(-4))-((3),(9))# #color(white)(3ulc)=((28),(-16))-((3),(9))=((25),(-25))# #ulc=1/3((25),(-25))=((25/3),(-25/3))# #rArrC=(25/3,-25/3)#