Points A and B are at #(4 ,1 )# and #(7 ,5 )#, respectively. Point A is rotated counterclockwise about the origin by #pi/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?
1 Answer
May 23, 2018
Explanation:
#"under a counterclockwise rotation about the origin of "pi/2#
#• " a point "(x,y)to(-y,x)#
#A(4,1)toA'(-1,4)" where A' is the image of A"#
#vec(CB)=color(red)(1/2)vec(CA')#
#"expressing in terms of position vectors gives"#
#ulb-ulc=1/2(ula'-ulc)#
#ulb-ulc=1/2ula'-1/2ulc#
#1/2ulc=ulb-1/2ula'#
#color(white)(1/2ulc)=((7),(5))-1/2((-1),(4))#
#color(white)(1/2ulc)=((7),(5))-((-1/2),(-2))=((15/2),(7))#
#ulc=1/2((15/2),(7))=((15/4),(7/2))#
#"thus "C=(15/4,7/2)#
