If point [tex]$P$[/tex] is [tex]$\frac{9}{11}$[/tex] of the distance from [tex]$M$[/tex] to [tex]$N$[/tex], what ratio does the point [tex]$P$[/tex] partition the directed line segment from [tex]$M$[/tex] to [tex]$N$[/tex] into?

A. [tex]$9:2$[/tex]
B. [tex]$9:9$[/tex]
C. [tex]$9:11$[/tex]
D. [tex]$9:13$[/tex]



Answer :

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 point [tex]\( P \)[/tex] is [tex]\(\frac{9}{11}\)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], follow these steps:

1. Understand the positioning of [tex]\( P \)[/tex]:
- The distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] can be considered as a whole unit, which is 1.
- Point [tex]\( P \)[/tex] is located at [tex]\(\frac{9}{11}\)[/tex] of this distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex].

2. Calculate the remaining distance:
- If point [tex]\( P \)[/tex] is [tex]\(\frac{9}{11}\)[/tex] of the way from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], the remaining distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is the fraction of the unit distance that's left, which can be calculated as:
[tex]\[ 1 - \frac{9}{11} = \frac{11}{11} - \frac{9}{11} = \frac{2}{11}. \][/tex]

3. Establish the ratio:
- The ratio in which [tex]\( P \)[/tex] partitions the segment can now be expressed as the ratio of the distance from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] to the distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex].
- This can be written as:
[tex]\[ \frac{\frac{9}{11}}{\frac{2}{11}}. \][/tex]

4. Simplify the ratio:
- The common denominator (11) can be cancelled out in the fraction, simplifying the ratio:
[tex]\[ \frac{9}{2}. \][/tex]
- Therefore, the ratio can be expressed as [tex]\( 9:2 \)[/tex].

So, the point [tex]\( P \)[/tex] divides the line segment [tex]\( MN \)[/tex] into the ratio [tex]\( 9:2 \)[/tex].

Hence, the correct answer is:
[tex]\[ \boxed{9:2} \][/tex]