Finding Coordinates Of Point N Midpoint Of ML

In geometry, understanding how to find the midpoint of a line segment is a fundamental skill. This article will guide you through the process of determining the coordinates of a point, specifically point N, which is the midpoint of segment ML. We'll start by establishing the given information: Point M is the midpoint of segment KL, and point N is the midpoint of segment ML. Point K is located at (-7, -6), and point L is located at (1, 10). Our goal is to find the coordinates of point N. This problem involves applying the midpoint formula twice, first to find the coordinates of point M and then to find the coordinates of point N. Let's dive in and break down the solution step by step.

Understanding the Midpoint Formula

The midpoint formula is a crucial tool for solving this problem. It states that the midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ has coordinates $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$. This formula essentially calculates the average of the x-coordinates and the average of the y-coordinates to find the point exactly in the middle of the segment. To effectively use the midpoint formula, it's essential to correctly identify the coordinates of the endpoints of the segment. The midpoint formula is derived from the concept of averaging the coordinates, ensuring that the resulting point is equidistant from both endpoints. This concept is widely used in various fields, including computer graphics, engineering, and navigation, making it a valuable skill to master.

Applying the Midpoint Formula to Find Point M

Our first step is to find the coordinates of point M, which is the midpoint of segment KL. We are given that point K is located at (-7, -6) and point L is located at (1, 10). Applying the midpoint formula, we have:

$M = \left( \frac{x_K + x_L}{2}, \frac{y_K + y_L}{2} \right)$

Substituting the coordinates of K and L, we get:

$M = \left( \frac{-7 + 1}{2}, \frac{-6 + 10}{2} \right)$

Simplifying the expressions, we find:

$M = \left( \frac{-6}{2}, \frac{4}{2} \right)$

$M = (-3, 2)$

Therefore, the coordinates of point M are (-3, 2). This result is crucial as it forms the basis for our next step in finding the coordinates of point N. Understanding this step is vital as it lays the groundwork for the subsequent calculations. The ability to accurately apply the midpoint formula in the first step ensures the correctness of the final answer. This methodical approach is essential for solving complex geometric problems.

Finding the Coordinates of Point N

Now that we have the coordinates of point M (-3, 2), we can proceed to find the coordinates of point N. Point N is the midpoint of segment ML, and we know the coordinates of point L (1, 10). Applying the midpoint formula again, this time for segment ML, we have:

$N = \left( \frac{x_M + x_L}{2}, \frac{y_M + y_L}{2} \right)$

Substituting the coordinates of M and L, we get:

$N = \left( \frac{-3 + 1}{2}, \frac{2 + 10}{2} \right)$

Simplifying the expressions, we find:

$N = \left( \frac{-2}{2}, \frac{12}{2} \right)$

$N = (-1, 6)$

Thus, the coordinates of point N are (-1, 6). This is our final answer. The process of finding the midpoint twice, first for KL and then for ML, highlights the importance of breaking down complex problems into simpler steps. Accurate calculations at each step are crucial for arriving at the correct solution. This methodical approach not only helps in solving geometric problems but also enhances problem-solving skills in general.

Verifying the Solution

To ensure the accuracy of our solution, it's always a good practice to verify the result. We found that point N has coordinates (-1, 6). Let's review the steps we took to arrive at this answer. First, we found the midpoint M of segment KL using the coordinates of K (-7, -6) and L (1, 10). We calculated M to be (-3, 2). Then, we used the coordinates of M (-3, 2) and L (1, 10) to find the midpoint N of segment ML. Our calculations yielded N as (-1, 6). Now, let’s intuitively check if this result makes sense. Point N should lie midway between M and L. Visually, on a coordinate plane, (-1, 6) does appear to be in the middle of (-3, 2) and (1, 10). While this is not a rigorous proof, it provides a sanity check and increases our confidence in the answer. Another way to verify is to calculate the distance between M and N and the distance between N and L. If these distances are equal, it confirms that N is indeed the midpoint of ML. This verification step underscores the importance of double-checking work to minimize errors and ensure accuracy in mathematical problem-solving.

Conclusion

In conclusion, we have successfully determined the coordinates of point N to be (-1, 6). This was achieved by applying the midpoint formula twice: first to find the midpoint M of segment KL and then to find the midpoint N of segment ML. The midpoint formula is a powerful tool in coordinate geometry, allowing us to find the exact center point of a line segment given its endpoints. The problem-solving approach used here highlights the importance of breaking down complex problems into smaller, manageable steps. Accurate calculations and careful application of formulas are crucial for arriving at the correct solution. Furthermore, verifying the solution adds an extra layer of confidence in the result. This method of finding midpoints has applications in various fields, making it a valuable skill to acquire. Mastering these concepts not only helps in solving geometric problems but also enhances analytical and problem-solving skills in general. Remember, practice is key to proficiency in mathematics, so continue to apply these techniques to various problems to solidify your understanding.

Final Answer

The coordinates of point N are (-1, 6), which corresponds to option C.