Question

rectangle abcd is the image of rectangle abcd after it has been translated according to the rule $t_{-4, 3}(x, y)$. which points are vertices of the pre - image, rectangle abcd? select four options. $\square (-1, -2)$ $\square (7, 1)$ $\square (-1, 7)$ $\square (-1, 1)$ $\square (7, -2)$

Explanation:

Step1: Identify image vertices

From the graph, the vertices of image $A'B'C'D'$ are:
$A'(-5, 4)$, $B'(3, 4)$, $C'(3, 1)$, $D'(-5, 1)$

Step2: Reverse translation rule

The translation rule is $T_{-4,3}(x,y)=(x-4,y+3)$. To find pre-image $(x,y)$ from image $(x',y')$, solve:
$x' = x - 4 \implies x = x' + 4$
$y' = y + 3 \implies y = y' - 3$

Step3: Calculate pre-image A

Substitute $A'(-5,4)$:
$x = -5 + 4 = -1$, $y = 4 - 3 = 1$
Pre-image $A(-1,1)$

Step4: Calculate pre-image B

Substitute $B'(3,4)$:
$x = 3 + 4 = 7$, $y = 4 - 3 = 1$
Pre-image $B(7,1)$

Step5: Calculate pre-image C

Substitute $C'(3,1)$:
$x = 3 + 4 = 7$, $y = 1 - 3 = -2$
Pre-image $C(7,-2)$

Step6: Calculate pre-image D

Substitute $D'(-5,1)$:
$x = -5 + 4 = -1$, $y = 1 - 3 = -2$
Pre-image $D(-1,-2)$

Answer:

$(-1, -2)$, $(7, 1)$, $(-1, 1)$, $(7, -2)$