If point $P$ is $\frac{9}{11}$ of the distance from $M$ to $N$, what ratio does the point $P$ partition the directed line segment from $M$ to $N$ into?
Answer (2)
Point P is 11 9 of the distance from M to N , meaning MP = 11 9 MN .
Calculate PN by subtracting MP from MN : PN = MN − MP = 11 2 MN .
Determine the ratio MP : PN by dividing MP by PN : PN MP = 11 2 MN 11 9 MN = 2 9 .
The ratio in which P partitions the directed line segment MN is 9 : 2 .
Explanation
Problem Analysis Let's analyze the problem. We are given that point P is 11 9 of the distance from M to N . This means that the length of the segment MP is 11 9 of the length of the segment MN . We need to find the ratio in which P partitions the directed line segment MN , which is the ratio of MP to PN .
Calculate PN We know that MP = 11 9 MN . To find the length of PN , we can subtract MP from MN :
PN = MN − MP = MN − 11 9 MN = 11 11 MN − 11 9 MN = 11 2 MN
Calculate the ratio MP:PN Now we have MP = 11 9 MN and PN = 11 2 MN . We can find the ratio MP : PN by dividing MP by PN :
PN MP = 11 2 MN 11 9 MN = 11 9 ⋅ 2 11 = 2 9
Final Answer Therefore, the ratio MP : PN is 9 : 2 .
Examples
In architecture, when designing a bridge or a building, engineers often need to divide a structural beam into specific ratios to ensure proper weight distribution and support. For instance, if a support column needs to be placed 11 9 of the way along a beam, this problem demonstrates how to calculate the ratio in which the column divides the beam, ensuring the structure's stability and integrity.
Point P partitions the directed line segment from M to N in the ratio of 9 : 2 . This is determined based on the length of segments MP and PN , where P is located 11 9 of the way from M to N . Therefore, the ratio can be calculated easily by evaluating the sizes of the segments created by point P .
;
