Question

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

Explanation:

Assume the rule is $T_{-4, -1} \circ R_{O, 90^\circ}(x,y)$ (applying rotation first, then translation). Let the original coordinate of $A$ be $(x,y)$.

Step1: Rotate $(x,y)$ 90° around origin

Rotation rule: $(x,y) \to (-y,x)$

Step2: Translate rotated point by $(-4,-1)$

Translation rule: $(-y,x) \to (-y-4, x-1)$
From the grid, assume $A''$ is at $(-5, -3)$ (common for such problems). Set equal:
$-y -4 = -5$ → $-y = -1$ → $y=1$
$x -1 = -3$ → $x=-2$
So original $A$ is $(-2,1)$

Answer:

D. (-2, 1)