To determine the ratio in which point [tex]\( P \)[/tex] partitions the line segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], let's break it down step-by-step.
Given:
- Point [tex]\( P \)[/tex] is [tex]\(\frac{9}{11}\)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex].
This means:
- The total distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] is divided such that [tex]\( P \)[/tex] is at a point that covers [tex]\(\frac{9}{11}\)[/tex] of this distance.
Now, to find the partition, consider the remaining part of the distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex]:
- The distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] would be the remaining fraction, which is [tex]\( 1 - \frac{9}{11} = \frac{2}{11} \)[/tex].
Thus, point [tex]\( P \)[/tex] divides the segment [tex]\( M \)[/tex] to [tex]\( N \)[/tex] into two parts:
- The part [tex]\( MP \)[/tex] is equivalent to [tex]\(\frac{9}{11}\)[/tex] of the total distance.
- The part [tex]\( PN \)[/tex] is equivalent to [tex]\(\frac{2}{11}\)[/tex] of the total distance.
Expressing this as a ratio, we have:
- The ratio of the segment [tex]\( MP \)[/tex] to the segment [tex]\( PN \)[/tex] is [tex]\( 9:2 \)[/tex].
Therefore, the correct ratio in which point [tex]\( P \)[/tex] partitions the directed line segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] is [tex]\( \boxed{9:2} \)[/tex].
