Question

the rule $r_{0,90^{circ}}circ t_{- 1,1}(x,y)$ is applied to $\triangle bcd$ to produce $\triangle bcd$. point $b$ of the final image is at $(-4,1)$. what are the coordinates of point $b$ on the pre - image? (2,3) (-2,-5) (5,0) (0,-5)

Explanation:

Step1: Analyze the transformation rules

The composition of transformations is a translation $T_{- 1,1}(x,y)=(x - 1,y + 1)$ followed by a $90^{\circ}$ counter - clockwise rotation $R_{0,90^{\circ}}(x,y)=(-y,x)$. Let the coordinates of point $B$ be $(x,y)$. After translation, the coordinates become $(x - 1,y + 1)$. After rotation, the coordinates of $B''$ are $(-(y + 1),x - 1)$.

Step2: Set up equations

We know that the coordinates of $B''$ are $(-4,1)$. So we have the following system of equations:
$- (y + 1)=-4$ and $x - 1 = 1$.
From $-(y + 1)=-4$, we get $y+1 = 4$, so $y=3$.
From $x - 1 = 1$, we get $x = 2$.

Answer:

A. $(2,3)$