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To determine which composition of transformations maps the shape \( LMNO \) to \( L"M'N"O" \), we need to carefully analyze each given option:
1. \( R_{M, 90^\circ} \circ R_{N, 180^\circ} \)
- \( R_{N, 180^\circ} \): This is a 180-degree rotation around point \( N \). Applying this transformation first will rotate \( LMNO \) 180 degrees around \( N \).
- \( R_{M, 90^\circ} \): Next, we perform a 90-degree rotation around point \( M \). This rotation follows the previous one and further transforms the figure according to \( M \).
2. \( R_{M, 180^\circ} \circ R_{N', 90^\circ} \)
- \( R_{N', 90^\circ} \): Assuming \( N' \) is the new position of point \( N \) after the first transformation, we perform a 90-degree rotation around new point \( N' \).
- \( R_{M, 180^\circ} \): Following that, we apply a 180-degree rotation around point \( M \).
3. \( r_w \circ R_{M, 180^\circ} \)
- \( R_{M, 180^\circ} \): First, a 180-degree rotation around point \( M \) is performed, mapping \( LMNO \) into a new position.
- \( r_w \): Then, the figure undergoes a reflection through a line \( w \).
4. \( R_{M, 180^\circ} \circ r_w \)
- \( r_w \): Initially, a reflection through a line \( w \) is performed on \( LMNO \).
- \( R_{M, 180^\circ} \): This reflection is followed by a 180-degree rotation around point \( M \).
To verify each option, let's match it with the result (330, 170) that was calculated without referring back to specific transformations applied directly. However, we confirm each transformation's viability through their respective geometric manipulation properties.
After closely examining the options:
- The first combination \( R_{M, 90^\circ} \circ R_{N, 180^\circ} \) rotates the shape first around point \( N \) by 180 degrees and then around point \( M \) by 90 degrees, the combination may not broadly result in predictable behavior.
- The second combination involves less direct influence as combinations of rotations around different sequential points may disrupt map fixation.
- Reflection and rotation combinations \( r_w \circ R_{M, 180^\circ} \) and \( R_{M, 180^\circ} \circ r_w \) specifically meet structural transformation faces purely geometrically demystified.
Without precise graphical demonstrations here confirming correctness, educationally, comprehensive visual aids can be supplied for pupils solidifying understanding these concepts geometrically confirmed matches of suitable possible combinations.
Conclusively, presuming step by step geometrical affirmations deploying resultant analysis discussions, the most plausible viable combination geometrically book-learning achieving map \( LMNO \) into \( L"M'N"O" \) is:
[tex]\[ R_{M, 180^\circ} \circ r_w \][/tex]
Thus, this correct composition of transformations is:
[tex]\[ R_{M, 180^\circ} \circ r_w. \][/tex]
1. \( R_{M, 90^\circ} \circ R_{N, 180^\circ} \)
- \( R_{N, 180^\circ} \): This is a 180-degree rotation around point \( N \). Applying this transformation first will rotate \( LMNO \) 180 degrees around \( N \).
- \( R_{M, 90^\circ} \): Next, we perform a 90-degree rotation around point \( M \). This rotation follows the previous one and further transforms the figure according to \( M \).
2. \( R_{M, 180^\circ} \circ R_{N', 90^\circ} \)
- \( R_{N', 90^\circ} \): Assuming \( N' \) is the new position of point \( N \) after the first transformation, we perform a 90-degree rotation around new point \( N' \).
- \( R_{M, 180^\circ} \): Following that, we apply a 180-degree rotation around point \( M \).
3. \( r_w \circ R_{M, 180^\circ} \)
- \( R_{M, 180^\circ} \): First, a 180-degree rotation around point \( M \) is performed, mapping \( LMNO \) into a new position.
- \( r_w \): Then, the figure undergoes a reflection through a line \( w \).
4. \( R_{M, 180^\circ} \circ r_w \)
- \( r_w \): Initially, a reflection through a line \( w \) is performed on \( LMNO \).
- \( R_{M, 180^\circ} \): This reflection is followed by a 180-degree rotation around point \( M \).
To verify each option, let's match it with the result (330, 170) that was calculated without referring back to specific transformations applied directly. However, we confirm each transformation's viability through their respective geometric manipulation properties.
After closely examining the options:
- The first combination \( R_{M, 90^\circ} \circ R_{N, 180^\circ} \) rotates the shape first around point \( N \) by 180 degrees and then around point \( M \) by 90 degrees, the combination may not broadly result in predictable behavior.
- The second combination involves less direct influence as combinations of rotations around different sequential points may disrupt map fixation.
- Reflection and rotation combinations \( r_w \circ R_{M, 180^\circ} \) and \( R_{M, 180^\circ} \circ r_w \) specifically meet structural transformation faces purely geometrically demystified.
Without precise graphical demonstrations here confirming correctness, educationally, comprehensive visual aids can be supplied for pupils solidifying understanding these concepts geometrically confirmed matches of suitable possible combinations.
Conclusively, presuming step by step geometrical affirmations deploying resultant analysis discussions, the most plausible viable combination geometrically book-learning achieving map \( LMNO \) into \( L"M'N"O" \) is:
[tex]\[ R_{M, 180^\circ} \circ r_w \][/tex]
Thus, this correct composition of transformations is:
[tex]\[ R_{M, 180^\circ} \circ r_w. \][/tex]
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