In The Coordinate Plane, Given The Directed Line Segment From K To N With Endpoints K(-6,-2) And N(8,3), Find The Coordinates Of Point L That Partitions The Segment In A Ratio Of 1:2, And Point M That Partitions The Segment In A Ratio Of 5:3.
Introduction
In coordinate geometry, understanding how to partition a directed line segment is a fundamental concept. This involves finding a point on the line segment that divides it into specific ratios. This article delves into the process of partitioning a directed line segment in the coordinate plane, using the example of segment KN with endpoints K(-6, -2) and N(8, 3), and points L and M that partition this segment in given ratios. We will explore the formulas and steps involved, providing a comprehensive guide for students and enthusiasts of mathematics.
Understanding Directed Line Segments
Before diving into the calculations, it's crucial to understand what a directed line segment is. A directed line segment is a line segment with a specific direction. In the coordinate plane, this direction is determined by the order of the endpoints. For instance, the directed line segment from K to N, denoted as KN, starts at point K and ends at point N. This is different from the segment NK, which starts at N and ends at K. The direction matters when we talk about partitioning the segment in a given ratio.
Partitioning a Directed Line Segment
When we partition a directed line segment, we are essentially finding a point on the line that divides it into two segments with a specific ratio. This ratio represents the proportional lengths of the two segments created by the partitioning point. The formula to find the coordinates of a point partitioning a line segment is derived from the concept of weighted averages. This formula ensures that the point we find lies on the line segment and divides it in the desired ratio.
The Partitioning Formula
The formula to find the coordinates of a point P(x, y) that partitions the directed line segment from A(x₁, y₁) to B(x₂, y₂) in the ratio m:n is given by:
- x = (mx₂ + nx₁) / (m + n)
- y = (my₂ + ny₁) / (m + n)
This formula is a cornerstone in coordinate geometry and is widely used in various applications, including computer graphics, geometric modeling, and more advanced mathematical concepts. Understanding this formula is crucial for mastering the art of partitioning line segments and solving related problems.
Point L Partitioning KN in a Ratio of 1:2
Let’s consider the first scenario: Point L partitions the directed segment KN in a ratio of 1:2. This means that the segment KL is one-third of the total length of KN, and the segment LN is two-thirds of the total length. We are given the coordinates of K(-6, -2) and N(8, 3). To find the coordinates of point L, we will apply the partitioning formula. This involves substituting the given values into the formula and performing the necessary calculations. The process may seem complex at first, but with practice, it becomes straightforward and intuitive.
Applying the Partitioning Formula
Using the formula, we have:
- x = (1 * 8 + 2 * (-6)) / (1 + 2)
- y = (1 * 3 + 2 * (-2)) / (1 + 2)
Now, let's simplify these expressions step by step. This will help us understand how the formula works and how to avoid common errors. The simplification process involves basic arithmetic operations, such as multiplication, addition, and division. Paying attention to each step ensures accuracy and clarity in the solution.
Calculating the x-coordinate of L
For the x-coordinate:
- x = (8 - 12) / 3
- x = -4 / 3
So, the x-coordinate of point L is -4/3. This value represents the horizontal position of point L on the coordinate plane. Understanding the significance of this value is crucial for visualizing the location of point L on the line segment KN. The x-coordinate, along with the y-coordinate, uniquely defines the position of point L.
Calculating the y-coordinate of L
For the y-coordinate:
- y = (3 - 4) / 3
- y = -1 / 3
Thus, the y-coordinate of point L is -1/3. This value represents the vertical position of point L on the coordinate plane. Similar to the x-coordinate, understanding the significance of the y-coordinate is crucial for accurately plotting the position of point L. Together, the x and y coordinates provide a complete picture of point L's location.
The Coordinates of Point L
Therefore, the coordinates of point L are (-4/3, -1/3). This is the point that divides the directed line segment KN in the ratio of 1:2. Visualizing this point on the coordinate plane helps solidify the understanding of the partitioning concept. The point L lies on the line segment KN, closer to point K than to point N, which aligns with the given ratio of 1:2.
Point M Partitioning KN in a Ratio of 5:3
Next, let's find the coordinates of point M, which partitions the directed segment KN in a ratio of 5:3. This means that the segment KM is five-eighths of the total length of KN, and the segment MN is three-eighths of the total length. Again, we will use the partitioning formula with the same endpoints K(-6, -2) and N(8, 3), but with a different ratio. This exercise reinforces the application of the partitioning formula and highlights how changing the ratio affects the position of the partitioning point.
Applying the Partitioning Formula for Point M
Using the formula, we have:
- x = (5 * 8 + 3 * (-6)) / (5 + 3)
- y = (5 * 3 + 3 * (-2)) / (5 + 3)
Now, we will simplify these expressions to find the coordinates of point M. The simplification process is similar to what we did for point L, but with different numerical values. This provides an opportunity to practice the arithmetic operations and reinforce the understanding of the formula.
Calculating the x-coordinate of M
For the x-coordinate:
- x = (40 - 18) / 8
- x = 22 / 8
- x = 11 / 4
So, the x-coordinate of point M is 11/4. This value represents the horizontal position of point M on the coordinate plane. Comparing this value with the x-coordinate of point L gives us a sense of the relative positions of the two points on the line segment KN.
Calculating the y-coordinate of M
For the y-coordinate:
- y = (15 - 6) / 8
- y = 9 / 8
Thus, the y-coordinate of point M is 9/8. This value represents the vertical position of point M on the coordinate plane. Similar to the x-coordinate, the y-coordinate provides crucial information about the location of point M. Together, the x and y coordinates uniquely define the position of point M.
The Coordinates of Point M
Therefore, the coordinates of point M are (11/4, 9/8). This is the point that divides the directed line segment KN in the ratio of 5:3. Point M is located further along the segment KN compared to point L, which is consistent with the larger ratio of 5:3. Visualizing the positions of points L and M on the coordinate plane helps solidify the understanding of how different ratios affect the partitioning of a line segment.
Conclusion
In conclusion, partitioning a directed line segment in the coordinate plane involves using a specific formula that takes into account the coordinates of the endpoints and the desired ratio. By applying this formula, we can accurately determine the coordinates of the point that divides the segment in the given ratio. In our example, we found that point L, which partitions KN in a ratio of 1:2, has coordinates (-4/3, -1/3), and point M, which partitions KN in a ratio of 5:3, has coordinates (11/4, 9/8). This process is fundamental in coordinate geometry and has various applications in mathematics and related fields. Mastering this concept is essential for a deeper understanding of geometric relationships and problem-solving in the coordinate plane.
This article has provided a step-by-step guide on how to partition a directed line segment, emphasizing the importance of understanding the formula and its application. By working through the example of segment KN and points L and M, readers can gain a solid grasp of this concept and confidently tackle similar problems. The ability to partition line segments is a valuable skill in mathematics, opening doors to more advanced topics and applications.