Question

which rule describes the composition of transformations that maps $\triangle abc$ to $\triangle abc$? $\bigcirc\\ r_{m}\circ r_{b,\\ 90^{\circ}}$ $\bigcirc\\ r_{b,\\ 90^{\circ}}\circ r_{m}$ $\bigcirc\\ r_{m}\circ r_{b,\\ 270^{\circ}}$ $\bigcirc\\ r_{b,\\ 270^{\circ}}\circ r_{m}$

Explanation:

Step1: Understand transformation notation

Composition notation $f \circ g$ means apply $g$ first, then $f$. $R_{g, \theta}$ = rotation about point $g$ by $\theta$; $r_m$ = reflection over line $m$.

Step2: Test option $R_{g, 90^\circ} \circ r_m$

First reflect $\triangle ABC$ over line $m$ to get $\triangle A''B''C''$. Then rotate $\triangle A''B''C''$ 90° clockwise (or 270° counterclockwise) about point $B'$ (matching $g$) to get $\triangle A'B'C''$.

Step3: Eliminate other options

• $r_m \circ R_{g, 90^\circ}$: Rotate first then reflect does not match the figure.
• $r_m \circ R_{g, 270^\circ}$: Rotate 270° then reflect gives incorrect orientation.
• $R_{g, 270^\circ} \circ r_m$: Reflect then 270° rotation gives reversed orientation.

Answer:

$R_{g,\ 90^\circ} \circ r_m$