Question

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

Explanation:

Step 1: Recall Transformations

• \( R_{M, \theta} \): Rotation about point \( M \) by \( \theta \) degrees.
• \( r_w \): Reflection over line \( w \).

Step 2: Analyze Each Option

• Option 1: \( R_{M, 90^\circ} \circ R_{N, 180^\circ} \)

Rotating about \( N \) first (180°) then about \( M \) (90°) won't align LMNO to L''M''N''O''.

• Option 2: \( R_{M, 180^\circ} \circ R_{N, 90^\circ} \)

Rotating about \( N \) (90°) then \( M \) (180°) also misaligns the figure.

• Option 3: \( r_w \circ R_{M, 180^\circ} \)

First, rotate LMNO 180° about \( M \) to get \( L'M'N'O' \) (matches the top blue rectangle). Then reflect over line \( w \) (horizontal line) to map to \( L''M''N''O'' \).

• Option 4: \( R_{M, 180^\circ} \circ r_w \)

Reflecting first then rotating 180° about \( M \) doesn't produce the correct alignment (as marked incorrect in the image).

Answer:

\( r_w \circ R_{M, 180^\circ} \) (the third option: \( r_{w} \circ R_{M, 180^\circ} \))