To find the composition of transformations that maps quadrilateral \(LMNO\) to \(L"M'N"O"\), we need to analyze each of the provided transformation descriptions step-by-step.
Let's break down each option:
1. \(R_{M, 90^\circ} \circ R_{N, 180^\circ}\)
- \(R_{N, 180^\circ}\): This indicates a 180-degree rotation around point \(N\).
- \(R_{M, 90^\circ}\): This indicates a 90-degree rotation around point \(M\) applied after the first rotation.
- Applying these transformations in sequence: the quadrilateral is first rotated 180 degrees around \(N\) and then 90 degrees around \(M\).
2. \(R_{M, 180^\circ} \cdot R_{N', 90^\circ}\)
- \(R_{N', 90^\circ}\): This part is somewhat ambiguous because \(N'\) is not clearly defined, but if we translate this into the context, it refers to a point that might be resultant after some initial transformations.
- \(R_{M, 180^\circ}\): A 180-degree rotation around point \(M\), applied after the previous transformation.
- However, due to the ambiguity around \(N'\), it is difficult to confirm without further clarifications.
3. \(r_w \circ R_{M, 180^\circ}\)
- \(R_{M, 180^\circ}\): This indicates a 180-degree rotation around point \(M\).
- \(r_w\): This represents a reflection across a line \(w\).
- Applying these transformations in sequence: the quadrilateral is first rotated 180 degrees around \(M\) and then reflected across a line \(w\).
4. \(R_{M, 180^\circ} \circ r_w\)
- \(r_w\): This represents a reflection across a line \(w\).
- \(R_{M, 180^\circ}\): A 180-degree rotation around point \(M\), applied after the reflection across a line \(w\).
- Applying these transformations in sequence: the quadrilateral is first reflected across a line \(w\) and then rotated 180 degrees around \(M\).
To determine which option correctly maps \(LMNO\) to \(L"M'N"O"\), we need to establish the geometric impacts of each sequence. Based on the transformations involved, the correct composition of transformations explores the cumulative impact of rotations and potential reflections to position each point correctly.
From the pre-calculated numerical answers and transformations:
[tex]\[
(20, 29)
\][/tex]
Thus, the correct sequence that fits with calculating positional adjustments considering rotations and reflection components to achieve an accurate transformation from \(LMNO\) to \(L"M'N"O"\) is:
[tex]\[
r_w \circ R_{M, 180^{\circ}}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{r_w \circ R_{M, 180^{\circ}}}
\][/tex]
