To determine the ratio in which point [tex]\( P \)[/tex] partitions the directed line segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], we first understand the given information. Point [tex]\( P \)[/tex] is [tex]\( \frac{4}{7} \)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex].
To find the ratio, let's break down the remaining part of the segment from [tex]\( P \)[/tex] to [tex]\( N \)[/tex].
Since [tex]\( P \)[/tex] is [tex]\( \frac{4}{7} \)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], the remaining part of the distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is:
[tex]\[ 1 - \frac{4}{7} = \frac{7}{7} - \frac{4}{7} = \frac{3}{7} \][/tex]
So, [tex]\( P \)[/tex] divides the line segment [tex]\( MN \)[/tex] into two parts:
- From [tex]\( M \)[/tex] to [tex]\( P \)[/tex], which is [tex]\( \frac{4}{7} \)[/tex] of the distance.
- From [tex]\( P \)[/tex] to [tex]\( N \)[/tex], which is [tex]\( \frac{3}{7} \)[/tex] of the distance.
The ratio of the distance [tex]\( M \)[/tex] to [tex]\( P \)[/tex] compared to the distance [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is thus:
[tex]\[ \frac{\frac{4}{7}}{\frac{3}{7}} = \frac{4}{3} \][/tex]
So, the ratio in which point [tex]\( P \)[/tex] partitions the line segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] is:
[tex]\[ 4 : 3 \][/tex]
Therefore, the correct answer is:
[tex]\[ 4:3 \][/tex]
