To determine the ratio in which point [tex]\( P \)[/tex] divides the directed line segment [tex]\( MN \)[/tex], considering that [tex]\( P \)[/tex] is [tex]\(\frac{4}{7}\)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], follow these steps:
1. Understand the given fraction: [tex]\( P \)[/tex] is [tex]\(\frac{4}{7}\)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex]. This means that if we denote the entire distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] as 7 units, [tex]\( P \)[/tex] is located 4 units away from [tex]\( M \)[/tex].
2. Calculate the remaining distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex]: If [tex]\( P \)[/tex] is 4 units from [tex]\( M \)[/tex] and the entire segment [tex]\( MN \)[/tex] is 7 units, then the distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is:
[tex]\[
7 - 4 = 3 \text{ units}
\][/tex]
3. Write the ratio in which [tex]\( P \)[/tex] divides [tex]\( MN \)[/tex]: Since [tex]\( P \)[/tex] divides [tex]\( MN \)[/tex] into two segments, one from [tex]\( M \)[/tex] to [tex]\( P \)[/tex], and the other from [tex]\( P \)[/tex] to [tex]\( N \)[/tex], and their lengths are 4 units and 3 units respectively, the ratio [tex]\( M \)[/tex] to [tex]\( P \)[/tex] : [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is:
[tex]\[
4:3
\][/tex]
Therefore, the point [tex]\( P \)[/tex] partitions the directed line segment [tex]\( M \)[/tex] to [tex]\( N \)[/tex] into a ratio of [tex]\( \boxed{4:3} \)[/tex].
