Question

which composition of transformations would map lmno to lmno?
$r_{m,90^{circ}}circ r_{n,180^{circ}}$
$r_{m,180^{circ}}circ r_{n,90^{circ}}$
$r_{w}circ r_{m,180^{circ}}$
$r_{m,180^{circ}}circ r_{w}$

Explanation:

Step1: Understand rotation and reflection notations

$R_{P,\theta}$ represents a rotation about point $P$ by an angle $\theta$, and $r_{w}$ represents a reflection across line $w$. Analyze each option by considering the effect of each transformation in the composition on the polygon LMNO.

Step2: Analyze option 1

For $R_{M,90^{\circ}}\circ R_{N,180^{\circ}}$, first rotating LMNO about $N$ by $180^{\circ}$ and then rotating the resulting figure about $M$ by $90^{\circ}$ does not map LMNO to L''M''N''O''.

Step3: Analyze option 2

For $R_{M,180^{\circ}}\circ R_{N,90^{\circ}}$, first rotating about $N$ by $90^{\circ}$ and then rotating the resulting figure about $M$ by $180^{\circ}$ does not map LMNO to L''M''N''O''.

Step4: Analyze option 3

For $r_{w}\circ R_{M,180^{\circ}}$, first rotating LMNO about $M$ by $180^{\circ}$ and then reflecting the resulting figure across line $w$ will map LMNO to L''M''N''O''.

Step5: Analyze option 4

For $R_{M,180^{\circ}}\circ r_{w}$, first reflecting across line $w$ and then rotating about $M$ by $180^{\circ}$ does not map LMNO to L''M''N''O''.

Answer:

$r_{w}\circ R_{M,180^{\circ}}$