Question

given △xyz with vertices x(-2,1), y(-5,-5), and z(-3,5), write the translation equivalent to the composition of transformations. the equation of line l is x = 2. (r_{l}circ r_{y - axis}) ((r_{l}circ r_{y - axis})(△xyz)=t_{squaresquare}(△xyz))

Explanation:

Step1: Reflection over y - axis rule

The rule for reflecting a point $(x,y)$ over the $y$-axis is $(x,y)\to(-x,y)$.
For point $X(-2,1)$, after reflection over the $y$-axis, $X_1=(2,1)$. For $Y(-5, - 5)$, $Y_1=(5,-5)$. For $Z(-3,5)$, $Z_1=(3,5)$.

Step2: Reflection over the line $x = 2$ rule

The distance between a point $(x,y)$ and the line $x = 2$ is $d=\vert x - 2\vert$. After reflection over the line $x=2$, the new $x$-coordinate is $x'=2+(2 - x)=4 - x$.
For $X_1=(2,1)$, after reflection over $x = 2$, $X_2=(2,1)$. For $Y_1=(5,-5)$, $Y_2=(4 - 5,-5)=(-1,-5)$. For $Z_1=(3,5)$, $Z_2=(4 - 3,5)=(1,5)$.

Step3: Find the translation vector

To get from the original points to the final - reflected points:
For $X(-2,1)$ to $X_2=(2,1)$, the change in $x$ is $2-(-2)=4$ and the change in $y$ is $1 - 1=0$.
For $Y(-5,-5)$ to $Y_2=(-1,-5)$, the change in $x$ is $-1-(-5)=4$ and the change in $y$ is $-5-(-5)=0$.
For $Z(-3,5)$ to $Z_2=(1,5)$, the change in $x$ is $1-(-3)=4$ and the change in $y$ is $5 - 5=0$.
The translation vector is $\langle4,0
angle$.

Answer:

$\langle4,0
angle$