1. The problem states that quadrilateral LMNP is similar to quadrilateral L'M'N'P'. We need to identify which equations correctly represent the ratios of corresponding sides to solve for missing lengths. 2. In similar polygons, corresponding sides are proportional. This means the ratio of one side in the first polygon to its corresponding side in the second polygon equals the ratio of another pair of corresponding sides. 3. The correct proportion equations must match corresponding sides. For quadrilaterals LMNP and L'M'N'P', the corresponding vertices and sides are: - L corresponds to L' - M corresponds to M' - N corresponds to N' - P corresponds to P' 4. Therefore, the sides correspond as follows: - LM corresponds to L'M' - MN corresponds to M'N' - NP corresponds to N'P' - PL corresponds to P'L' 5. Let's analyze each equation: - Equation 1: $\frac{N'P}{M'U} = \frac{NP}{ML}$ - $N'P$ and $M'U$ are not corresponding sides (also $M'U$ is not a side in the original quadrilateral), so this is incorrect. - Equation 2: $\frac{NP}{N'P'} = \frac{ML}{M'U}$ - $NP$ corresponds to $N'P'$, correct. - $ML$ corresponds to $M'U$? $ML$ is side $LM$ reversed, but $M'U$ is not a side in the quadrilateral, so incorrect. - Equation 3: $\frac{MN}{M'N'} = \frac{PL}{P'L'}$ - $MN$ corresponds to $M'N'$, correct. - $PL$ corresponds to $P'L'$, correct. - This equation is valid. - Equation 4: $\frac{MN}{PL} = \frac{M'N'}{P'L'}$ - $MN$ and $PL$ are sides in the same quadrilateral, not corresponding sides. - $M'N'$ and $P'L'$ are sides in the other quadrilateral. - This does not represent corresponding sides, so incorrect. - Equation 5: $\frac{PL}{MN'} = \frac{PL}{MN}$ - $MN'$ is not a side in the quadrilateral. - This equation is invalid. 6. Conclusion: Only Equation 3 correctly represents the proportion of corresponding sides in similar quadrilaterals. Final answer: The equation $\frac{MN}{M'N'} = \frac{PL}{P'L'}$ can be used to solve for missing lengths.