Question
If point P is Four-sevenths of the distance from M to N, what ratio does the point P partition the directed line segment from M to N into?
4:1
4:3
4:7
4:10
4:1
4:3
4:7
4:10
Answers
GPT-4o mini
To find the ratio in which point P partitions the directed line segment from M to N, we need to understand that if P is four-sevenths of the distance from M to N, then this means that P is closer to M than to N.
If we denote the entire segment MN as having a length of 7 units in this context (for simplification, let's assume the total distance MN = 1):
- The distance from M to P would then be \(\frac{4}{7}\), which leaves the distance from P to N as \(\frac{3}{7}\).
Therefore, the ratio of the distances from M to P and from P to N is:
\[
\text{Ratio} = \frac{MP}{PN} = \frac{\frac{4}{7}}{\frac{3}{7}} = \frac{4}{3}
\]
Thus, point P partitions the directed line segment from M to N in the ratio of \(4:3\).
Therefore, the correct answer is:
**4:3**
If we denote the entire segment MN as having a length of 7 units in this context (for simplification, let's assume the total distance MN = 1):
- The distance from M to P would then be \(\frac{4}{7}\), which leaves the distance from P to N as \(\frac{3}{7}\).
Therefore, the ratio of the distances from M to P and from P to N is:
\[
\text{Ratio} = \frac{MP}{PN} = \frac{\frac{4}{7}}{\frac{3}{7}} = \frac{4}{3}
\]
Thus, point P partitions the directed line segment from M to N in the ratio of \(4:3\).
Therefore, the correct answer is:
**4:3**