If point [tex]$P$[/tex] is [tex]$\frac{9}{11}$[/tex] of the distance from [tex]$M$[/tex] to [tex]$N$[/tex], what ratio does the point [tex]$P$[/tex] partition the directed line segment from [tex]$M$[/tex] to [tex]$N$[/tex] into?
Answer (1)
Point P divides the segment MN such that MP is 11 9 of MN .
Express PN as MN − MP , which gives PN = 11 2 MN .
Calculate the ratio PN MP as 11 2 MN 11 9 MN .
Simplify the ratio to find the answer: 9 : 2 .
Explanation
Analyze the problem 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 want to find the ratio in which P partitions the directed line segment MN , which is the ratio MP : PN .
Express PN in terms of MN and MP We know that MP = 11 9 MN . Since P lies on the segment MN , we have MP + PN = MN . We can express PN in terms of MN and MP as follows: PN = MN − MP
Substitute the value of MP Substitute MP = 11 9 MN into the equation for PN :
PN = MN − 11 9 MN = 11 11 MN − 11 9 MN = 11 2 MN
Calculate the ratio MP:PN Now we can find the ratio MP : PN :
PN MP = 11 2 MN 11 9 MN = 11 9 ÷ 11 2 = 11 9 × 2 11 = 2 9
State the final ratio The ratio MP : PN is 2 9 , which can be written as 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.
