To determine the ratio in which point [tex]\( P \)[/tex] partitions the directed line segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], given that [tex]\( P \)[/tex] is [tex]\(\frac{9}{11}\)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex]:
1. Understanding the partition:
- Given [tex]\( P \)[/tex] is [tex]\(\frac{9}{11}\)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], we need to understand how many segments or parts from [tex]\( M \)[/tex] this represents relative to the total distance between [tex]\( M \)[/tex] and [tex]\( N \)[/tex].
2. Segment interpretation:
- The line can be thought of as being divided into 11 equal parts.
- Out of these 11 parts, point [tex]\( P \)[/tex] is located 9 parts from [tex]\( M \)[/tex].
3. Remaining segments calculation:
- Since [tex]\( P \)[/tex] is 9 parts from [tex]\( M \)[/tex], to find the remaining segments to reach [tex]\( N \)[/tex], we subtract the 9 parts from the total of 11 parts.
[tex]\[
\text{Remaining segments to } N = 11 - 9 = 2
\][/tex]
4. Ratio formation:
- The distance from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] is 9 parts.
- The distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is 2 parts.
- Therefore, the ratio that point [tex]\( P \)[/tex] partitions the line segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] into is [tex]\( 9:2 \)[/tex].
So the correct ratio is [tex]\( 9:2 \)[/tex].
