Answer:
[tex]A'= (2,-1)[/tex] and [tex]R'=(4,-2)[/tex] under this translation.
Step-by-step explanation:
A translation in [tex]R^{2}[/tex] is a mapping T from [tex]R^{2}[/tex] to [tex]R^{2}[/tex] defined by [tex]T(x,y) = (x + v_1,y+v_2)[/tex], where [tex]v=(v_1,v_2)[/tex] is a fixed vector in [tex]R^{2}[/tex].
From the problem we know that [tex]T(2,4)=(3,2)[/tex], so we need to find the values [tex]v_1[/tex] and [tex]v_2[/tex] such that [tex]T(2,4) = (2 + v_1,4+v_2)=(3,2)[/tex], so [tex]3=2 + v_1[/tex] and [tex]4+v_2=2[/tex], thus [tex]v_1=1[/tex] and [tex]v_2=2[/tex].
Then [tex]T(x,y) = (x + 1,y-2)[/tex] and
[tex]T(1,1)=(1+1,1-2)=(2,-1)=A'[/tex]
[tex]T(3,0)=(3+1,0-2)=(4,-2)=R'[/tex]
Therefore [tex]A'= (2,-1)[/tex] and [tex]R'=(4,-2)[/tex]. The triangles CAR and C'Q'R' are shown in the figure below.
