Let's analyze the problem step-by-step:
1. Understanding the given information: We are told that point [tex]\( P \)[/tex] is located [tex]\(\frac{9}{11}\)[/tex] of the distance from point [tex]\( M \)[/tex] to point [tex]\( N \)[/tex].
2. Partition of the line segment: This means that the distance from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] is [tex]\(\frac{9}{11}\)[/tex] of the total distance [tex]\( MN \)[/tex].
3. Remaining part of the line segment: The remaining distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] would, therefore, be the rest of the line segment, which is [tex]\(1 - \frac{9}{11} = \frac{2}{11}\)[/tex] of the total distance [tex]\( MN \)[/tex].
4. Ratio of partition: The ratio that point [tex]\( P \)[/tex] partitions the line segment [tex]\( MN \)[/tex] into is therefore the distance from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] compared to the distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex].
- The distance from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] is [tex]\(\frac{9}{11}\)[/tex] of [tex]\( MN \)[/tex].
- The distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is [tex]\(\frac{2}{11}\)[/tex] of [tex]\( MN \)[/tex].
5. Expressing the ratio: To express the partition ratio as a simple ratio, we consider the distances as whole numbers. Since [tex]\(\frac{9}{11}\)[/tex] and [tex]\(\frac{2}{11}\)[/tex] share a common denominator, we can drop the denominator for the purpose of expressing the ratio:
- The distance from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] = 9 parts
- The distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] = 2 parts
6. Final Ratio: Thus, the ratio that point [tex]\( P \)[/tex] partitions the line segment [tex]\( MN \)[/tex] into is [tex]\(9 : 2\)[/tex].
So, the correct answer is [tex]\( \boxed{9:2} \)[/tex].
