Question

which rule describes the composition of transformations that maps △abc to △abc?
○ r_m ∘ r_{b, 90°}
○ r_{b, 90°} ∘ r_m
○ r_m ∘ r_{b, 270°}
○ r_{b, 270°} ∘ r_m

Explanation:

Step1: Analyze rotation first

First, observe that to get from the initial $\triangle ABC$ to an intermediate - step triangle, a rotation is likely. A $90^{\circ}$ counter - clockwise rotation about a point (say $B$) is a common transformation. The notation for a rotation of $90^{\circ}$ counter - clockwise about a point $B$ is $R_{B,90^{\circ}}$.

Step2: Analyze reflection

After the rotation, a reflection across the line $m$ is needed to map the intermediate - step triangle to $\triangle A'B'C'$. The notation for a reflection across a line $m$ is $r_{m}$. The composition of transformations is written such that the transformation on the right is performed first and the one on the left is performed second. So the composition of rotation first and then reflection is $r_{m}\circ R_{B,90^{\circ}}$.

Answer:

$r_{m}\circ R_{B,90^{\circ}}$