If point $P$ is $\frac{4}{7}$ of the distance from $M$ to $N$, what ratio does the point $P$ partition the directed line segment from $M$ to N into?
A. $4: 1$
B. $4: 3$
C. $4: 7$
D. $4: 10
Answer (2)
Point P is 7 4 of the distance from M to N .
Distance from M to P is 7 4 d .
Distance from P to N is 7 3 d .
The ratio is 4 : 3 .
Explanation
Problem Analysis Let's analyze the problem. We are given that point P is 7 4 of the distance from point M to point N . We need to find the ratio in which point P divides the directed line segment MN .
Calculate distances Let the distance from M to N be d . Then the distance from M to P is 7 4 d . The distance from P to N is the total distance d minus the distance from M to P , which is d − 7 4 d = 7 7 d − 7 4 d = 7 3 d .
Determine the ratio The ratio in which P partitions the directed line segment MN is the ratio of the distance from M to P to the distance from P to N , which is MP : PN = 7 4 d : 7 3 d . To simplify this ratio, we can divide both sides by 7 1 d . This gives us 4 : 3 .
Final Answer Therefore, the point P partitions the directed line segment from M to N in the ratio 4 : 3 .
Examples
Imagine you're baking a cake and the recipe says to add flour and sugar in a specific ratio. If the recipe says to use 7 4 of the total dry ingredients as flour, then the remaining 7 3 would be sugar. This means the ratio of flour to sugar is 4 : 3 . Understanding ratios helps you maintain the correct proportions in your baking, ensuring your cake turns out delicious!
Point P partitions the directed line segment from M to N in the ratio of 4 : 3 . This means that out of a total distance, 4 parts are from M to P , while 3 parts are from P to N . Therefore, the correct answer is option B: 4 : 3 .
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