Question

ΔCAR has coordinates C (2, 4), A (1, 1), and R (3, 0). A translation maps point C to C' (3, 2). Find the coordinates of A' and R' under this translation. (6 points) A' (4, −2); R' (2, −1) A' (−2, 2); R' (2, −2) A' (2, −1); R' (4, −2) A' (−1, 0); R' (−2, 2).

Answer 1

`A'= (2,-1)` and
`R'=(4,-2)` under this translation.

Explanation:

A translation in
`R^(2)` is a mapping T from
`R^(2)` to
`R^(2)` defined by
`T(x,y) = (x + v_1,y+v_2)`, where
`v=(v_1,v_2)` is a fixed vector in
`R^(2)`.

From the problem we know that
`T(2,4)=(3,2)`, so we need to find the values
`v_1` and
`v_2` such that
`T(2,4) = (2 + v_1,4+v_2)=(3,2)`, so
`3=2 + v_1` and
`4+v_2=2`, thus
`v_1=1` and
`v_2=2`.

Then
`T(x,y) = (x + 1,y-2)` and

`T(1,1)=(1+1,1-2)=(2,-1)=A'`

`T(3,0)=(3+1,0-2)=(4,-2)=R'`

Therefore
`A'= (2,-1)` and
`R'=(4,-2)`. The triangles CAR and C'Q'R' are shown in the figure below.