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Magazine Chapter 7: Problem 92
Find the midpoint of the line segment whose endpoints are given. See Example 7 $$ (-2,5),(-1,6) $$
Short Answer
Expert verified
The midpoint is (-1.5, 5.5).
Step by step solution
01
Understand the Midpoint Formula
The midpoint of a line segment in the coordinate plane having endpoints \(x_1, y_1\) and \(x_2, y_2\) is given by the formula: \[ \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
02
Identify the Coordinates
From the given endpoints, identify \(x_1, y_1\) as \(-2, 5\) and \(x_2, y_2\) as \(-1, 6\).
03
Calculate the Midpoint's X-Coordinate
Apply the formula for the midpoint's x-coordinate: \[ \frac{x_1 + x_2}{2} = \frac{-2 + (-1)}{2} = \frac{-3}{2} \] The x-coordinate of the midpoint is \(-1.5\).
04
Calculate the Midpoint's Y-Coordinate
Apply the formula for the midpoint's y-coordinate: \[ \frac{y_1 + y_2}{2} = \frac{5 + 6}{2} = \frac{11}{2} \] The y-coordinate of the midpoint is \(5.5\).
05
Write the Midpoint Coordinates
Combine the x and y coordinates found in the previous steps: The midpoint of the line segment is \(-1.5, 5.5\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Coordinate Geometry
Coordinate Geometry is a fascinating area of mathematics that combines geometry and algebra to determine the placements and relationships between various points, lines, and shapes within a plane. Here, algebraic equations can be used to define geometric properties much more precisely:
- It aids in understanding how points interact and connect within a defined space or plane.
- This field involves many transformations and calculations, often applied in real-world scenarios such as geographical mappings, architectural designs, and even in computer graphics.
Line Segment
A line segment is a fundamental concept in geometry. It consists of two endpoints and all the points between them on a given straight line. Unlike a line that stretches infinitely in both directions, a line segment is defined by its finite length.
- A simple way to understand a line segment is to think of it as a finite "piece" of a straight line.
- This concept is practical when you want to measure or find a part of something, like cutting a plank to a specific length.
- It's significant in coordinate geometry because it's often necessary to work with finite distances between points rather than continuous, infinite lines.
Coordinate Plane
The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. By using two perpendicular number lines, known as the x-axis and y-axis, the plane provides a grid that is essential for maintaining a systematic approach to organize and locate points.
- Each point in this plane is captured by a pair of numerical values (coordinates) which show its exact position on the grid.
- It allows anyone to effortlessly visualize and analyze relationships between various geometric elements.
- The coordinate plane serves multiple functions, from solving mathematical problems to representing data graphically in various fields such as economics or physics.
Midpoint Calculation
Midpoint Calculation is a straightforward but powerful concept in coordinate geometry, especially beneficial when you want to find the central point of a line segment defined by two endpoints. This is achieved using the Midpoint Formula.
- The formula is: \( \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \)
- It provides a precise mean location between any two points on a coordinate plane, making it useful in tasks requiring symmetry or balance.
- For instance, if you have endpoints \((-2, 5)\) and \((-1, 6)\), calculating the midpoint gives you: \((-1.5, 5.5)\).
