Alright, let's dive into this problem step by step.
1. Understanding the Problem:
- We are given that point [tex]\( P \)[/tex] is located at [tex]\(\frac{4}{7}\)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex].
- We need to determine the ratio in which point [tex]\( P \)[/tex] partitions the directed line segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex].
2. Interpreting the Distance:
- If [tex]\( P \)[/tex] is [tex]\(\frac{4}{7}\)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], this means the remaining distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] will be:
[tex]\[
1 - \frac{4}{7} = \frac{3}{7}.
\][/tex]
3. Setting Up the Ratio:
- The ratio of the distances from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] and [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is [tex]\(\frac{4}{7}\)[/tex] (for [tex]\( M \)[/tex] to [tex]\( P \)[/tex]) and [tex]\(\frac{3}{7}\)[/tex] (for [tex]\( P \)[/tex] to [tex]\( N \)[/tex]).
- In ratio terms, this yields the ratio:
[tex]\[
4 : 3.
\][/tex]
4. Answer:
- Thus, the point [tex]\( P \)[/tex] partitions the directed line segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] in the ratio of [tex]\( 4 : 3 \)[/tex].
Therefore, the correct answer is:
[tex]\[
4 : 3
\][/tex]
