Question

which rule describes the composition of transformations that maps △bcd to △b\c\d\?
t_{5, - 6} \circ r_{y=-x}(x,y)
r_{y=-x} \circ t_{5, - 6}(x,y)
t_{6, - 5} \circ r_{y - axis}(x,y)
r_{y - axis} \circ t_{6, - 5}(x,y)

Explanation:

Step1: Analyze translation

First, observe the horizontal and vertical shifts. Comparing the original triangle $\triangle BCD$ and the intermediate - transformed triangle (after the first transformation), we can see that to get from $\triangle BCD$ to the triangle before the reflection, we need to translate. If we consider a general point $(x,y)$ of $\triangle BCD$, we can see that it is translated 5 units to the right and 6 units down. The translation rule $T_{a,b}(x,y)=(x + a,y + b)$ gives $T_{5,-6}(x,y)=(x + 5,y-6)$.

Step2: Analyze reflection

Then, observe the reflection. The final triangle $\triangle B''C''D''$ is obtained by reflecting the triangle after translation across the line $y=-x$. The reflection rule $r_{y = - x}(x,y)=(-y,-x)$. The composition of transformations is first translation and then reflection, which is written as $r_{y=-x}\circ T_{5,-6}(x,y)$.

Answer:

$r_{y = - x}\circ T_{5,-6}(x,y)$