Midpoint Of Line Segment GH A Step-by-Step Solution

In the realm of coordinate geometry, a fundamental concept is determining the midpoint of a line segment. The midpoint, as the name suggests, is the point that divides the segment into two equal parts. This article will delve into the process of calculating the midpoint, specifically focusing on the line segment GH with endpoints G(14, 3) and H(10, -6). We will explore the midpoint formula, its application, and the step-by-step calculations involved, ensuring a comprehensive understanding of this essential geometric concept.

Understanding the Midpoint Formula

The midpoint formula is a powerful tool for finding the coordinates of the midpoint of a line segment in a coordinate plane. It leverages the coordinates of the endpoints to pinpoint the exact center of the segment. The formula is derived from the concept of averaging the x-coordinates and the y-coordinates of the endpoints. Given two points, say extit{G}(x₁, y₁) and extit{H}(x₂, y₂), the midpoint extit{M} of the line segment GH is calculated as follows:

$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

This formula essentially finds the average of the x-coordinates and the average of the y-coordinates, giving us the coordinates of the point that lies exactly in the middle of the segment. The midpoint formula is a cornerstone of coordinate geometry, enabling us to solve a wide range of problems related to lines, segments, and geometric figures. Its simplicity and effectiveness make it an indispensable tool in mathematical analysis and problem-solving.

Applying the Midpoint Formula to Segment GH

To find the midpoint of the line segment GH, where G(14, 3) and H(10, -6), we will apply the midpoint formula. First, we identify the coordinates of the endpoints: x₁ = 14, y₁ = 3, x₂ = 10, and y₂ = -6. Now, we substitute these values into the formula:

$$M = \left(\frac{14 + 10}{2}, \frac{3 + (-6)}{2}\right)$$

Next, we perform the arithmetic operations within the parentheses. We add the x-coordinates (14 + 10) and divide by 2, and we add the y-coordinates (3 + (-6)) and divide by 2. This gives us:

$$M = \left(\frac{24}{2}, \frac{-3}{2}\right)$$

Finally, we simplify the fractions to obtain the coordinates of the midpoint:

$$M = \left(12, -\frac{3}{2}\right)$$

Therefore, the midpoint of the line segment GH is (12, -3/2). This point lies exactly halfway between G and H, dividing the segment into two equal parts. Understanding this application of the midpoint formula is crucial for solving various geometric problems and understanding spatial relationships in coordinate geometry.

Step-by-Step Calculation

Let's break down the calculation into a step-by-step process to ensure clarity. Given the endpoints G(14, 3) and H(10, -6), we want to find the midpoint M.

• Step 1: Identify the coordinates.

  • G(x₁, y₁) = (14, 3)
  • H(x₂, y₂) = (10, -6)

• Step 2: Apply the midpoint formula.

  • $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

• Step 3: Substitute the coordinates into the formula.

  • $$M = \left(\frac{14 + 10}{2}, \frac{3 + (-6)}{2}\right)$$

• Step 4: Simplify the expressions inside the parentheses.

  • $$M = \left(\frac{24}{2}, \frac{-3}{2}\right)$$

• Step 5: Perform the division to find the coordinates of the midpoint.

  • $$M = \left(12, -\frac{3}{2}\right)$$

This step-by-step breakdown clearly illustrates how the midpoint formula is applied to find the midpoint of a line segment. Each step is a logical progression, ensuring accurate calculation and understanding of the process. This methodical approach is beneficial for solving similar problems in coordinate geometry.

Analyzing the Options

Now that we have calculated the midpoint of GH to be (12, -3/2), let's analyze the given options to identify the correct answer. We have the following options:

A. (6, -15)

B. (-2, -9/2)

C. (12, -3/2)

D. (24, -3)

E. (18, 12)

Comparing our calculated midpoint (12, -3/2) with the options, we can clearly see that option C, (12, -3/2), matches our result. The other options have different x and y coordinates, making them incorrect. This analysis reinforces the importance of accurate calculation and comparison when solving mathematical problems. By systematically working through the problem and verifying the answer against the given options, we can confidently arrive at the correct solution.

Why Other Options Are Incorrect

To further solidify our understanding, let's examine why the other options are incorrect. This will help us identify common errors and reinforce the correct application of the midpoint formula.

• Option A: (6, -15)

  • This option is incorrect because the x-coordinate (6) is not the average of the x-coordinates of G and H (14 and 10). Similarly, the y-coordinate (-15) is not the average of the y-coordinates of G and H (3 and -6). It seems like there was a miscalculation in averaging the coordinates.

• Option B: (-2, -9/2)

  • This option is also incorrect. The x-coordinate (-2) is not the average of 14 and 10. The y-coordinate (-9/2) is also not the correct average of 3 and -6. This suggests a misunderstanding of the midpoint formula or an error in the arithmetic.

• Option D: (24, -3)

  • This option's x-coordinate (24) appears to be the sum of the x-coordinates of G and H (14 + 10), rather than their average. The y-coordinate (-3) is double the correct y-coordinate of the midpoint. This indicates a confusion between summing the coordinates and averaging them.

• Option E: (18, 12)

  • This option is incorrect as both the x and y coordinates do not represent the average values. The x-coordinate (18) is not the average of 14 and 10, and the y-coordinate (12) is significantly different from the average of 3 and -6.

By understanding why these options are incorrect, we gain a deeper appreciation for the correct application of the midpoint formula and the importance of accurate calculations. This exercise helps in avoiding common pitfalls and strengthens our problem-solving skills.

Conclusion

In conclusion, finding the midpoint of a line segment is a fundamental concept in coordinate geometry. The midpoint formula, which averages the x-coordinates and y-coordinates of the endpoints, provides a straightforward method for determining the midpoint. By applying this formula to the line segment GH with endpoints G(14, 3) and H(10, -6), we accurately calculated the midpoint to be (12, -3/2). This result aligns with option C, confirming it as the correct answer.

We also analyzed the other options, identifying the errors in their calculations, which reinforced our understanding of the midpoint formula and its correct application. This comprehensive exploration not only answers the specific question but also equips us with the knowledge and skills to tackle similar problems in coordinate geometry with confidence and accuracy. The midpoint formula is an essential tool in mathematical analysis, enabling us to solve a wide range of problems related to lines, segments, and geometric figures.

By mastering the concept of the midpoint and its calculation, we enhance our ability to analyze geometric relationships and solve complex problems in mathematics and related fields. The midpoint formula serves as a cornerstone in our mathematical toolkit, providing a reliable method for finding the center of a line segment and furthering our understanding of spatial geometry.