Question

1. $overline{rs}: r(2, - 1)$ and $s(6, - 5)$

translation: along $langle3,4
angle$
reflection: in $x = 2$

Explanation:

Step1: Apply translation

First, apply the translation $\langle3,4
angle$ to points $R(2, - 1)$ and $S(6,-5)$. For a point $(x,y)$ translated by $\langle a,b
angle$, the new - point is $(x + a,y + b)$.
For point $R(2,-1)$:
$R'=(2 + 3,-1 + 4)=(5,3)$
For point $S(6,-5)$:
$S'=(6 + 3,-5 + 4)=(9,-1)$

Step2: Apply reflection

The formula for reflecting a point $(x,y)$ over the vertical line $x = c$ is $(2c - x,y)$. Here $c = 2$.
For point $R'(5,3)$:
$R''=(2\times2 - 5,3)=(-1,3)$
For point $S'(9,-1)$:
$S''=(2\times2 - 9,-1)=(-5,-1)$

Answer:

The new coordinates of $R$ are $(-1,3)$ and the new coordinates of $S$ are $(-5,-1)$