To determine the ratio in which point [tex]\( P \)[/tex] partitions the directed line segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] when [tex]\( P \)[/tex] is [tex]\(\frac{4}{7}\)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], we can follow these steps:
1. Understanding the given fraction: Point [tex]\( P \)[/tex] is [tex]\(\frac{4}{7}\)[/tex] of the distance from point [tex]\( M \)[/tex] to point [tex]\( N \)[/tex]. This means that [tex]\( P \)[/tex] divides the segment [tex]\( M \)[/tex] to [tex]\( N \)[/tex] such that [tex]\( P \)[/tex] is closer to [tex]\( M \)[/tex].
2. Calculate the remaining distance: If [tex]\( P \)[/tex] is [tex]\(\frac{4}{7}\)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], the remaining distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is:
[tex]\[
1 - \frac{4}{7} = \frac{3}{7}
\][/tex]
3. Setting up the ratio: The ratio of the segment from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] compared to the segment from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] is given by:
[tex]\[
\frac{\frac{4}{7}}{\frac{3}{7}}
\][/tex]
4. Simplifying the ratio:
[tex]\[
\frac{\frac{4}{7}}{\frac{3}{7}} = \frac{4}{3}
\][/tex]
So, point [tex]\( P \)[/tex] partitions the segment [tex]\( M \)[/tex] to [tex]\( N \)[/tex] into a ratio of [tex]\( 4:3 \)[/tex].
Therefore, the point [tex]\( P \)[/tex] divides the directed line segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] in the ratio [tex]\( 4:3 \)[/tex].
The correct answer is:
[tex]\[
\boxed{4:3}
\][/tex]
