To solve the given problem, we need to determine how the point [tex]\( P \)[/tex] partitions the directed line segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] given that [tex]\( P \)[/tex] is at [tex]\(\frac{4}{7}\)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex].
### Step-by-Step Solution:
1. Understanding the Ratio:
- We are given that the distance from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] is [tex]\(\frac{4}{7}\)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex].
- This implies the total distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] can be considered as being divided into 7 equal parts.
2. Identifying the Partitions:
- Since [tex]\( P \)[/tex] is [tex]\( \frac{4}{7} \)[/tex] of the way from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], the distance from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] consists of 4 of these parts.
3. Remaining Distance:
- To find the distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex], we subtract the distance from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] from the total distance.
- The total number of parts is 7, and since 4 parts are from [tex]\( M \)[/tex] to [tex]\( P \)[/tex], the remaining parts from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] are [tex]\( 7 - 4 = 3 \)[/tex] parts.
4. Forming the Ratio:
- Hence, the ratio of the distances [tex]\( M \)[/tex] to [tex]\( P \)[/tex] and [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is [tex]\( 4 \)[/tex] parts to [tex]\( 3 \)[/tex] parts, respectively.
5. Final Ratio:
- Therefore, the ratio that the point [tex]\( P \)[/tex] partitions the directed line segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] is [tex]\( \boxed{4:3} \)[/tex].
The correct answer is [tex]\( \boxed{4:3} \)[/tex].
