Question

the rule $r_{y = x} \circ t_{4, 0}(x, y)$ is applied to trapezoid abcd to produce the final image abcd. which ordered pairs name the coordinates of vertices of the pre - image, trapezoid abcd? select two options. $\square (-1, 0)$ $\square (-1, -5)$ $\square (1, 1)$ $\square (7, 0)$ $\square (7, -5)$

Explanation:

Step1: List final image coordinates

From the graph:
$A''(-4, 5)$, $B''(-2, 5)$, $C''(0, 3)$, $D''(-5, 3)$

Step2: Reverse reflection $r_{y=x}$

The reflection $r_{y=x}(x,y)=(y,x)$, so its inverse is also $r_{y=x}(x,y)=(y,x)$. Apply to final image:
$A'(5, -4)$, $B'(5, -2)$, $C'(3, 0)$, $D'(3, -5)$

Step3: Reverse translation $T_{4,0}$

The translation $T_{4,0}(x,y)=(x+4,y)$, so inverse is $T_{-4,0}(x,y)=(x-4,y)$. Apply to $A',B',C',D'$:
$A(5-4, -4)=(1, -4)$
$B(5-4, -2)=(1, -2)$
$C(3-4, 0)=(-1, 0)$
$D(3-4, -5)=(-1, -5)$

Step4: Match with options

Compare pre-image coordinates to given options.

Answer:

$\boldsymbol{(-1, 0)}$, $\boldsymbol{(-1, -5)}$