Finding The Midpoint Of Line Segment GH A Step By Step Guide

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In coordinate geometry, determining the midpoint of a line segment is a fundamental concept with numerous applications. This article delves into the process of finding the midpoint of a line segment, using the specific example of line segment GH with endpoints G(10,1) and H(3,5). We will explore the midpoint formula, its derivation, and step-by-step application to solve the problem. Understanding this concept is crucial for various mathematical problems, including geometry, calculus, and linear algebra. Let's embark on this journey to master the art of finding midpoints.

The midpoint formula is a powerful tool that allows us to calculate the coordinates of the point exactly halfway between two given points in a coordinate plane. This formula is derived from the concept of averaging the x-coordinates and the y-coordinates of the two endpoints. The resulting point represents the geometric center of the line segment connecting the two endpoints. Mastering the midpoint formula is essential for solving a wide range of geometric problems, including finding the center of a circle, determining the intersection of medians in a triangle, and various other applications in coordinate geometry and beyond. This formula serves as a cornerstone for understanding spatial relationships and is a valuable asset in mathematical problem-solving. The formula provides a straightforward method for pinpointing the central location within a line segment, making it a fundamental tool in the study of geometry and related fields.

Understanding the Midpoint Formula

The midpoint formula is a fundamental concept in coordinate geometry, providing a straightforward method to determine the exact center of a line segment. Given two points, say G(x1,y1){G(x_1, y_1)} and H(x2,y2){H(x_2, y_2)}, the midpoint M{M} is calculated by averaging the x-coordinates and the y-coordinates separately. Mathematically, this is expressed as:

M=(x1+x22,y1+y22){ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) }

This formula essentially finds the average position between the two points in both the horizontal and vertical directions. To understand the intuition behind this formula, consider the x-coordinates. The average of the x-coordinates, x1+x22{\frac{x_1 + x_2}{2}}, gives the x-coordinate of the point that is equidistant from both x1{x_1} and x2{x_2}. A similar logic applies to the y-coordinates. By averaging both coordinates, we locate the point that is exactly in the middle of the line segment. The midpoint formula is not just a mathematical equation; it's a representation of geometric symmetry and balance. It allows us to precisely locate the center of a line segment, which is crucial in various geometric constructions and proofs. For instance, it's used to find the center of a circle when given the endpoints of a diameter, or to determine the intersection point of the medians of a triangle. This formula's simplicity and versatility make it a cornerstone of coordinate geometry, enabling us to solve a multitude of problems with ease and precision. The midpoint formula's elegance lies in its ability to translate a geometric concept – the center of a line segment – into a simple algebraic expression, making it accessible and applicable in various mathematical contexts.

Applying the Midpoint Formula to Find the Midpoint of GH

To find the midpoint of the line segment GH‾{\overline{GH}}, where G{G} is at (10,1){(10, 1)} and H{H} is at (3,5){(3, 5)}, we will apply the midpoint formula. This formula, as discussed earlier, involves averaging the x-coordinates and the y-coordinates of the endpoints. In this case, x1=10{x_1 = 10}, y1=1{y_1 = 1}, x2=3{x_2 = 3}, and y2=5{y_2 = 5}. Substituting these values into the midpoint formula gives us:

M=(10+32,1+52){ M = \left( \frac{10 + 3}{2}, \frac{1 + 5}{2} \right) }

Now, we perform the arithmetic operations to simplify the expression. First, we add the x-coordinates: 10+3=13{10 + 3 = 13}. Then, we add the y-coordinates: 1+5=6{1 + 5 = 6}. This gives us:

{ M = \left( \frac{13}{2}, rac{6}{2} \right) }

Next, we divide each sum by 2 to find the average. 132{\frac{13}{2}} remains as a fraction, which is equal to 6.5. 62{\frac{6}{2}} simplifies to 3. Therefore, the midpoint M{M} is:

M=(132,3){ M = \left( \frac{13}{2}, 3 \right) }

This result indicates that the midpoint of the line segment GH‾{\overline{GH}} is located at the coordinates (132,3){(\frac{13}{2}, 3)}. This point is exactly halfway between the points G{G} and H{H} in the coordinate plane. The application of the midpoint formula provides a precise and efficient method to determine this central location. This process highlights the practical utility of the midpoint formula in solving geometric problems, demonstrating its ability to transform coordinate values into meaningful spatial information. The calculated midpoint serves as a critical reference point for further geometric analysis, and its accurate determination is essential for various applications in mathematics and related fields.

Step-by-Step Solution

Let's break down the solution into clear, manageable steps to ensure a thorough understanding of the process. This step-by-step approach will not only help in solving this particular problem but also in applying the midpoint formula to other similar scenarios.

Step 1: Identify the Coordinates of the Endpoints

The first crucial step is to correctly identify the coordinates of the endpoints of the line segment. In our case, the endpoints are given as G(10,1){G(10, 1)} and H(3,5){H(3, 5)}. This means that the coordinates of point G{G} are x1=10{x_1 = 10} and y1=1{y_1 = 1}, and the coordinates of point H{H} are x2=3{x_2 = 3} and y2=5{y_2 = 5}. Ensuring the correct identification of these coordinates is paramount, as they form the foundation for the subsequent calculations. A simple mistake at this stage can lead to an incorrect midpoint, so it's essential to double-check these values before proceeding further. This initial step sets the stage for the accurate application of the midpoint formula, and its careful execution is key to achieving the correct result. The clarity and precision at this stage significantly contribute to the overall accuracy of the solution.

Step 2: Apply the Midpoint Formula

Now that we have identified the coordinates of the endpoints, the next step is to apply the midpoint formula. As we discussed earlier, the midpoint formula is given by:

M=(x1+x22,y1+y22){ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) }

This formula provides a systematic way to calculate the coordinates of the midpoint by averaging the x-coordinates and the y-coordinates of the endpoints. Substituting the values we identified in Step 1, we get:

M=(10+32,1+52){ M = \left( \frac{10 + 3}{2}, \frac{1 + 5}{2} \right) }

This substitution is a crucial step in translating the geometric problem into an algebraic expression that can be easily solved. The midpoint formula acts as a bridge between the coordinate plane and the realm of arithmetic, allowing us to precisely determine the center of the line segment through simple calculations. This step is where the abstract concept of a midpoint becomes a tangible set of coordinates, highlighting the power of mathematical formulas to represent and solve geometric problems. The correct application of the midpoint formula is essential for obtaining the accurate coordinates of the midpoint, making it a key step in the overall solution process.

Step 3: Simplify the Expression

The expression obtained in the previous step needs to be simplified to arrive at the final coordinates of the midpoint. This involves performing the arithmetic operations within the parentheses. Let's start with the x-coordinate:

10+32=132{ \frac{10 + 3}{2} = \frac{13}{2} }

Next, we simplify the y-coordinate:

1+52=62=3{ \frac{1 + 5}{2} = \frac{6}{2} = 3 }

These simplifications involve basic addition and division, but they are crucial for expressing the coordinates in their simplest form. The simplification process transforms the fractional expressions into easily interpretable values, allowing us to pinpoint the midpoint's exact location on the coordinate plane. This step is not just about calculation; it's about clarity and precision. A simplified expression is easier to work with and less prone to errors in subsequent calculations or interpretations. The goal is to present the midpoint coordinates in a clear and concise manner, making it readily applicable in further geometric analysis or problem-solving scenarios. The simplified expression is the key to unlocking the final answer and providing a clear representation of the midpoint's position.

Step 4: State the Midpoint Coordinates

After simplifying the expression, we arrive at the final coordinates of the midpoint. Based on our calculations, the midpoint M{M} is located at:

M=(132,3){ M = \left( \frac{13}{2}, 3 \right) }

This means that the x-coordinate of the midpoint is 132{\frac{13}{2}} (or 6.5), and the y-coordinate is 3. Stating the coordinates clearly and concisely is the final step in solving the problem. It's important to present the answer in a format that is easy to understand and interpret. The coordinates (132,3){(\frac{13}{2}, 3)} represent the exact center of the line segment GH‾{\overline{GH}}, providing a precise location in the coordinate plane. This final statement of the midpoint coordinates is the culmination of the entire solution process. It encapsulates the application of the midpoint formula, the simplification of the expression, and the accurate determination of the central point between the two given endpoints. This clear and concise answer serves as a valuable piece of information for further geometric analysis or related problem-solving activities, solidifying the importance of each step in the solution process.

Conclusion

In conclusion, we have successfully determined the midpoint of the line segment GH‾{\overline{GH}} with endpoints G(10,1){G(10, 1)} and H(3,5){H(3, 5)} using the midpoint formula. By applying the formula and simplifying the resulting expression, we found the midpoint to be (132,3){(\frac{13}{2}, 3)}. This exercise demonstrates the power and utility of the midpoint formula in coordinate geometry. Understanding and applying this formula is crucial for solving a variety of geometric problems. The step-by-step approach we followed provides a clear and systematic method for finding the midpoint of any line segment given its endpoints. This process not only reinforces the concept of the midpoint but also enhances problem-solving skills in geometry. The ability to accurately determine midpoints is fundamental for further exploration of geometric concepts and their applications in various fields. The midpoint formula serves as a cornerstone of geometric analysis, enabling us to precisely locate the center of a line segment and utilize this information for a wide range of mathematical and practical purposes. Mastering this concept is an essential step in building a strong foundation in geometry and related disciplines.

Therefore, the correct answer is C. (132,3){\left(\frac{13}{2}, 3\right)}