Question

square abcd is the final image after the rule $t_{-4,-1}circ r_{0,90^{circ}}(x,y)$ was applied to square abcd. what are the coordinates of vertex a of square abcd? (-1, -6) (-1, -2) (-1, 6) (-2, 1)

Explanation:

Step1: Analyze the transformation rules

The transformation $T_{- 4,-1}\circ R_{0,90}$ means a 90 - degree rotation about the origin followed by a translation 4 units left and 1 unit down. To find the original coordinates, we need to reverse the operations. First, reverse the translation (move 4 units right and 1 unit up) and then reverse the 90 - degree rotation (do a 270 - degree rotation about the origin).
Let's assume the coordinates of $A''$ are $(-4,-3)$.

Step2: Reverse the translation

If we reverse the translation $T_{-4,-1}$, we add 4 to the $x$ - coordinate and 1 to the $y$ - coordinate of $A''$. Let $(x',y')$ be the coordinates after reversing the translation. If $(x_{A''},y_{A''})=(-4,-3)$, then $x'=x_{A''}+4=-4 + 4=0$ and $y'=y_{A''}+1=-3 + 1=-2$.

Step3: Reverse the 90 - degree rotation

The rule for a 90 - degree rotation about the origin is $(x,y)\to(-y,x)$. To reverse a 90 - degree rotation (i.e., do a 270 - degree rotation), the rule is $(x,y)\to(y, - x)$.
If the coordinates after reversing the translation are $(0,-2)$, then after reversing the 90 - degree rotation, the original coordinates of $A$ are $(-2,1)$.

Answer:

$(-2,1)$