Question

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

Explanation:

Step1: Analyze rotation and reflection

First, observe the orientation - change of the polygon. A rotation of 180 - degree about a point changes the orientation of the figure. A reflection across a line also changes the position and orientation.

Step2: Check each option

For $R_{M,90^{\circ}}\circ R_{N,180^{\circ}}$, the order and angles of rotation are not correct for the given transformation. For $R_{M,180^{\circ}}\circ R_{N,90^{\circ}}$, the combination of rotations about different points does not match the transformation. For $r_{w}\circ R_{M,180^{\circ}}$, first rotating 180 - degrees about point $M$ and then reflecting across line $w$ gives the correct mapping of the polygon $LMNO$ to $L''M''N''O''$. For $R_{M,180^{\circ}}\circ r_{w}$, the order of reflection and rotation is wrong for the given transformation.

Answer:

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