Question

using structure

1. quadrilateral $defg$ has vertices $d(-1, 2)$, $e(-2, 0)$, $f(-1, -1)$, and $g(1, 3)$. a translation maps quadrilateral $defg$ to quadrilateral $defg$. the image of $d$ is $d(-2, -2)$. what are the coordinates of $e$, $f$, and $g$?

Explanation:

Step1: Find translation rule

Let translation be $(x,y)\to(x+a,y+b)$.
For $D(-1,2)\to D'(-2,-2)$:

$$\begin{cases}-1+a=-2\\2+b=-2\end{cases}$$

Solve: $a=-2-(-1)=-1$, $b=-2-2=-4$.
Translation: $(x,y)\to(x-1,y-4)$

Step2: Calculate $E'$ coordinates

$E(-2,0)$: $x'=-2-1=-3$, $y'=0-4=-4$
$E'=(-3,-4)$

Step3: Calculate $F'$ coordinates

$F(-1,-1)$: $x'=-1-1=-2$, $y'=-1-4=-5$
$F'=(-2,-5)$

Step4: Calculate $G'$ coordinates

$G(1,3)$: $x'=1-1=0$, $y'=3-4=-1$
$G'=(0,-1)$

Answer:

$E'(-3, -4)$, $F'(-2, -5)$, $G'(0, -1)$