Sure! Let's break down the problem step-by-step:
1. Understanding the Given Information: We are given that point [tex]\( P \)[/tex] is [tex]\( \frac{9}{11} \)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex]. This means if the total distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] is divided into 11 equal parts, [tex]\( P \)[/tex] is located 9 parts away from [tex]\( M \)[/tex], and the remaining distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is the remaining 2 parts out of 11.
2. Total Distance Division: If the whole segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] is considered to be 11 units (since the fraction provided is out of 11), we can set up the distances as follows:
- [tex]\( MP \)[/tex] (distance from [tex]\( M \)[/tex] to [tex]\( P \)[/tex]) = 9 units.
- [tex]\( PN \)[/tex] (distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex]) = [tex]\( 11 - 9 \)[/tex] units = 2 units.
3. Determining the Ratio: The point [tex]\( P \)[/tex] therefore divides the segment [tex]\( MN \)[/tex] into two parts:
- One part is 9 units (from [tex]\( M \)[/tex] to [tex]\( P \)[/tex]).
- The other part is 2 units (from [tex]\( P \)[/tex] to [tex]\( N \)[/tex]).
4. Ratio of Segments: Consequently, the ratio in which point [tex]\( P \)[/tex] divides the segment [tex]\( MN \)[/tex] is the ratio of the distances from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] and from [tex]\( P \)[/tex] to [tex]\( N \)[/tex]:
- This ratio is [tex]\( 9:2 \)[/tex].
Therefore, point [tex]\( P \)[/tex] divides the directed line segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] into a [tex]\( 9:2 \)[/tex] ratio.
The correct answer is:
[tex]\[ \boxed{9:2} \][/tex]
