Question

square a\b\c\d\ 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 transformation rules

The rule $T_{- 4,1}\circ R_{0,90^{\circ}}(x,y)$ means a 90 - degree counter - clockwise rotation about the origin followed by a translation 4 units left and 1 unit up. To find the original point, we need to reverse these operations. First, reverse the translation (move 4 units right and 1 unit down) and then reverse the rotation (a 90 - degree clockwise rotation about the origin).

Step2: Assume coordinates of $A''$

Let's assume the coordinates of $A''$ (from the graph) are (-5,0). Reverse the translation: add 4 to the x - coordinate and subtract 1 from the y - coordinate. So we get $(-5 + 4,0 - 1)=(-1,-1)$.

Step3: Reverse the rotation

For a 90 - degree clockwise rotation about the origin, if a point $(x,y)$ is rotated, the new point $(x',y')$ has the transformation $(x,y)\to(y, - x)$. For the point $(-1,-1)$, after the 90 - degree clockwise rotation, we get $(-1,-(-1))=(-1,-6)$.

Answer:

(-1, -6)