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 rule $T_{- 4,-1}\circ R_{0,90^{\circ}}(x,y)$ means a $90 -$ degree counter - clockwise rotation about the origin ($R_{0,90^{\circ}}$) followed by a translation ($T_{-4,-1}$). The rotation rule for a $90 -$ degree counter - clockwise rotation about the origin is $(x,y)\to(-y,x)$, and the translation rule $T_{-4,-1}$ is $(x,y)\to(x - 4,y - 1)$. Let the original coordinates of point $A$ be $(x,y)$, after rotation it becomes $(-y,x)$ and after translation it becomes $(-y-4,x - 1)$.

Step2: Find the coordinates of $A'$ from the graph

From the graph, the coordinates of $A'$ are $(-5,5)$. So we set up the following system of equations:

$$\begin{cases}-y-4=-5\\x - 1 = 5\end{cases}$$

Step3: Solve the system of equations

For the first equation $-y-4=-5$, add 4 to both sides: $-y=-5 + 4=-1$, then $y = 1$. For the second equation $x-1 = 5$, add 1 to both sides: $x=5 + 1=6$. The original coordinates of $A$ before the rotation and translation are $(-1,6)$ (because when we reverse the operations, we first reverse the translation and then the rotation).

Answer:

$(-1,6)$