Introduction

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Understanding the concept of the midpoint is crucial in coordinate geometry. This article aims to provide a clear and concise explanation of how to calculate the midpoint of a line segment, using the given example and extending it to cover broader applications.

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Calculating the Midpoint: A Step-by-Step Guide

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What is the Midpoint?

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The midpoint of a line segment is the point that divides the segment into two equal parts. Mathematically, given two points (A37 and B7-3), the midpoint can be found using the midpoint formula.

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Formula and Steps

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Given two points (A(x_1, y_1)) and (B(x_2, y_2)), the midpoint (M) of the line segment AB can be calculated as:

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Midpoint Formula:

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(M left( frac{x_1 x_2}{2}, frac{y_1 y_2}{2} right))

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Step-by-Step Example

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Using the example provided, the points (A(3, 7)) and (B(-3, -7)).

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1. Calculate the Average of the x-coordinates

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[x_{mid} frac{3 (-3)}{2} frac{0}{2} 0]

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2. Calculate the Average of the y-coordinates

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[y_{mid} frac{7 (-7)}{2} frac{0}{2} 0]

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3. Determine the Midpoint

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The midpoint (M) of the line segment through points (A(3, 7)) and (B(-3, -7)) is:

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[M (0, 0)]

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Context and Importance

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The midpoint is not only a fundamental concept in geometry but also plays a key role in various fields such as physics, engineering, and computer graphics. Understanding how to find the midpoint is crucial for solving a wide range of problems involving line segments.

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Related Concepts and Applications

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1. Distance Formula:

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The distance between two points (A(x_1, y_1)) and (B(x_2, y_2)) is given by:

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[d sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2}]

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The midpoint helps in dividing the segment into two equal halves, which can be useful for calculating distances and poses.

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2. Parallel Lines and Perpendicular Bisectors:

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The midpoint is crucial for determining the perpendicular bisector of a line segment, which is a line that intersects the segment at its midpoint and forms a 90-degree angle with the segment.

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Understanding the midpoint can help in constructing and analyzing geometric figures, proving theorems, and solving practical problems.

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Conclusion

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Calculating the midpoint is a foundational skill in coordinate geometry. It provides a simple yet powerful tool for analyzing and solving problems involving line segments. By mastering this concept, you can tackle more complex geometric and practical challenges with confidence.