To determine whether point [tex]\( P \)[/tex] is closer to [tex]\( A \)[/tex] or [tex]\( B \)[/tex] when it partitions the line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] in the ratio [tex]\( 3:4 \)[/tex], we need to analyze the division of the segment based on the given ratio.
1. Understanding the Ratio:
- The ratio [tex]\( 3:4 \)[/tex] means that the segment from [tex]\( A \)[/tex] to [tex]\( P \)[/tex] is 3 parts, and the segment from [tex]\( P \)[/tex] to [tex]\( B \)[/tex] is 4 parts.
2. Total Parts:
- Summing up the parts, we get the total number of parts as [tex]\( 3 + 4 = 7 \)[/tex].
3. Distance Partition:
- The distance [tex]\( AP \)[/tex] (from [tex]\( A \)[/tex] to [tex]\( P \)[/tex]) will be [tex]\( \frac{3}{7} \)[/tex] of the total length of the segment.
- The distance [tex]\( PB \)[/tex] (from [tex]\( P \)[/tex] to [tex]\( B \)[/tex]) will be [tex]\( \frac{4}{7} \)[/tex] of the total length of the segment.
4. Comparison of Distances:
- Since [tex]\( \frac{4}{7} \)[/tex] (approximately 0.571) is greater than [tex]\( \frac{3}{7} \)[/tex] (approximately 0.429), it is evident that [tex]\( P \)[/tex] is closer to [tex]\( B \)[/tex] than to [tex]\( A \)[/tex].
5. Conclusion:
- Point [tex]\( P \)[/tex] will be closer to [tex]\( B \)[/tex] because it will be [tex]\( \frac{3}{7} \)[/tex] of the distance from [tex]\( B \)[/tex] to [tex]\( A \)[/tex], and [tex]\( \frac{3}{7} \)[/tex] (approximately 0.429) is less than [tex]\( \frac{4}{7} \)[/tex] (approximately 0.571).
Thus, the correct answer is:
[tex]\[ P \text{ will be closer to } B \text{ because it will be } \frac{3}{7} \text{ the distance from } B \text{ to } A. \][/tex]
