Answer: 4:3.
Step-by-step explanation:
Given: Point P is [tex]\dfrac{4}{7}[/tex] of the distance from M to N.
To find: The ratio in which the point P partition the directed line segment from M to N.
If Point P is between points M and N, then the ratio can be written as
[tex]\dfrac{MP}{MN}=\dfrac{MP}{MP+PN}[/tex]
As per given,
[tex]\dfrac{MP}{MP+PN}=\dfrac{4}{7}\\\\\Rightarrow\ \dfrac{MP+PN}{MP}=\dfrac{7}{4}\\\\\Rightarrow\ \dfrac{MP}{MP}+\dfrac{PN}{MP}=\dfrac{7}{4}\\\\\Rightarrow\ -1+\dfrac{PN}{MP}=\dfrac{7}{4}\\\\\Rightarrow\ \dfrac{PN}{MP}=\dfrac{7}{4}-1=\dfrac{7-4}{4}=\dfrac{3}{4}\\\\\Rightarrow\ \dfrac{PN}{MP}=\dfrac{3}{4}\ \ \or\ \dfrac{MP}{PN}=\dfrac{4}{3}[/tex]
Hence, P partition the directed line segment from M to N in 4:3.
