Ratio Partition Of A Line Segment If Point P Is 4/7 Of The Distance From M To N

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Introduction

In the realm of geometry, understanding how points partition line segments is a fundamental concept. This article delves into the problem of determining the ratio in which a point P divides a directed line segment MN, given that the distance MP is 4/7 of the total distance MN. This problem not only reinforces basic geometric principles but also highlights the importance of ratios and proportions in solving mathematical problems. We will explore the problem step by step, providing a clear and comprehensive solution that is easy to understand.

Problem Statement

The problem at hand states that point P lies on the directed line segment MN such that the distance from M to P (MP) is 4/7 of the total distance from M to N (MN). Our objective is to determine the ratio in which point P partitions the directed line segment MN. This means we need to find the ratio of the length of the segment MP to the length of the segment PN. The problem presents us with multiple-choice options, and through careful analysis, we will identify the correct ratio.

Understanding Directed Line Segments and Ratios

Before diving into the solution, it’s essential to understand the concepts of directed line segments and ratios. A directed line segment is a line segment with a specific direction, meaning the order of the endpoints matters. In our case, MN implies a segment starting at point M and ending at point N. The length of a line segment is the distance between its endpoints.

A ratio is a comparison of two quantities by division. If we say the ratio of a to b is x:y, it means that a/b = x/y. In the context of partitioning a line segment, the ratio represents how the segment is divided into two parts by a point. For instance, if a point P divides a line segment AB in the ratio 2:3, it means that the length of AP is 2/5 of the total length AB, and the length of PB is 3/5 of the total length AB.

Detailed Solution

To solve the problem, let's break down the given information and use it to find the required ratio.

  1. Given Information: We know that MP = (4/7)MN. This tells us that the distance from M to P is 4/7 of the total distance from M to N.
  2. Finding PN: To determine the ratio in which P partitions MN, we need to find the length of the segment PN. Since P lies on MN, we can express PN as the difference between the total length MN and the length MP. Mathematically, PN = MN - MP.
  3. Substituting MP: We know MP = (4/7)MN, so we substitute this into the equation for PN: PN = MN - (4/7)MN.
  4. Simplifying PN: To simplify the expression, we can rewrite MN as (7/7)MN. Thus, PN = (7/7)MN - (4/7)MN = (3/7)MN. This tells us that the distance from P to N is 3/7 of the total distance from M to N.
  5. Determining the Ratio: Now that we have MP and PN in terms of MN, we can find the ratio MP:PN. We have MP = (4/7)MN and PN = (3/7)MN. The ratio MP:PN is therefore ((4/7)MN) : ((3/7)MN).
  6. Simplifying the Ratio: To simplify the ratio, we can divide both sides by (1/7)MN, which gives us 4:3. Therefore, the point P partitions the directed line segment MN in the ratio 4:3.

Analyzing the Options

Now, let’s compare our solution with the given options:

A. 4:1 B. 4:3 C. 4:7 D. 4:10

Our calculated ratio of 4:3 matches option B. Therefore, the correct answer is B.

Why the Other Options are Incorrect

It's important to understand why the other options are incorrect to reinforce our understanding of the problem.

  • Option A (4:1): This would imply that PN is only 1/4 the length of MP. Given that MP is 4/7 of MN, this would mean PN is much shorter than it actually is.
  • Option C (4:7): This ratio might seem plausible because the fraction 4/7 is given in the problem. However, it represents the ratio of MP to the entire segment MN, not MP to PN.
  • Option D (4:10): This ratio is incorrect because it does not accurately represent the relationship between MP and PN based on the given information. If we simplify 4:10, we get 2:5, which doesn't align with the condition that MP is 4/7 of MN.

Key Takeaways

From this problem, we can glean several key takeaways:

  • Understanding Fractions: The ability to work with fractions and understand their proportional relationships is crucial in solving geometric problems.
  • Partitioning Line Segments: When a point partitions a line segment, the sum of the lengths of the resulting segments equals the length of the original segment.
  • Ratios and Proportions: Ratios provide a way to compare quantities, and understanding how to manipulate ratios is essential in geometry.
  • Step-by-Step Approach: Breaking down a complex problem into smaller, manageable steps can make it easier to solve.

Real-World Applications

The concept of partitioning line segments and understanding ratios is not just limited to theoretical mathematics. It has numerous real-world applications in various fields, including:

  • Architecture and Engineering: Architects and engineers use ratios to design structures and ensure that components are proportionally sized.
  • Navigation: Navigators use ratios to calculate distances and plan routes, whether on land, sea, or air.
  • Computer Graphics: In computer graphics, ratios are used to scale objects and create realistic perspectives.
  • Cartography: Mapmakers use ratios to represent distances on a map relative to actual distances on the ground.

Conclusion

In conclusion, we have successfully determined that point P partitions the directed line segment MN in the ratio 4:3. This was achieved by carefully analyzing the given information, understanding the concepts of directed line segments and ratios, and applying logical reasoning. The problem highlights the importance of proportional relationships and provides a solid foundation for tackling more complex geometric problems. By understanding these fundamental concepts, one can appreciate their wide-ranging applications in both theoretical and practical contexts. Remember, the key to mastering geometry lies in a thorough understanding of basic principles and a step-by-step approach to problem-solving. This problem serves as an excellent example of how breaking down a complex problem into smaller, manageable parts can lead to a clear and accurate solution. The ability to apply these skills is invaluable not only in mathematics but also in various real-world scenarios where proportional reasoning is essential.

This detailed exploration should provide a comprehensive understanding of the problem and its solution, reinforcing the importance of ratios and proportional reasoning in geometry.