Points A and B are at #(2 ,5 )# and #(6 ,2 )#, respectively. Point A is rotated counterclockwise about the origin by #(3pi)/2 # 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?

2 Answers
Jun 8, 2018

#color(red)("Coordinates of " C (7,6)#

Explanation:

#A(2,5), B(6,2), "rotation " #(3pi)/2#, "dilation factor" 1/2#

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

#A(2,5) rarr A' (5,-2)#

#vec (BC) = (1/2) vec(A'C)#

#b - c = (1/2)a' - (1/2)c#

#(1/2)c = -(1/2)a' + b#

#(1/2)C((x),(y)) = -(1/2)((5),(-2)) + ((6),(2)) = ((7/2),(3))#

#color(red)("Coordinates of " 2 *C ((7/2),3) = C(7,6)#

Jun 8, 2018

#C=(7,6)#

Explanation:

#"under a counterclockwise rotation about the origin of "(3pi)/2#

#• " a point "(x,y)to(y,-x)#

#A(2,5)toA'(5,-2)" where A' is the image of A"#

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

#ulb-ulc=1/2(ula'-ulc)#

#ulb-ulc=1/2ula'-1/2ulc#

#1/2ulc=ulb-1/2ula'#

#color(white)(1/2ulc)=((6),(2))-1/2((5),(-2))#

#color(white)(1/2ulc)=((6),(2))-((5/2),(-1))=((7/2),(3))#

#ulc=2((7/2),(3))=((7),(6))#

#rArrC=(7,6)#