Problem 92 - Intermediate Algebra

Question

Find the midpoint of the line segment whose endpoints are given. See Example 7 $$ (-2,5),(-1,6) $$

Step-by-Step Solution

Answer

The midpoint is (-1.5, 5.5).

Step 1: 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: \[ \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]

Step 2: Identify the Coordinates

From the given endpoints, identify \(x_1, y_1\) as \(-2, 5\) and \(x_2, y_2\) as \(-1, 6\).

Step 3: 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 \(-1.5\).

Step 4: 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 \(5.5\).

Step 5: Write the Midpoint Coordinates

Combine the x and y coordinates found in the previous steps: The midpoint of the line segment is \(-1.5, 5.5\).

Key Concepts

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.

The magic of coordinate geometry lies in its ability to describe complex shapes and lines using numerical coordinates. So, if you have ever wondered how we can definitively locate, measure, or map objects within a space, coordinate geometry is your answer!

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.

Line segments can be easily visualized and manipulated on a coordinate plane, transforming abstract mathematical ideas into tangible forms.

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.

Whether plotting simple line segments or more intricate shapes, the coordinate plane is indispensable for visualizing how different elements relate to one another.

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: \((-1.5, 5.5)\).

This calculation helps in segmenting designs in architecture and maps precisely, ensuring equidistant placement of elements, reflecting harmony and coordination in visual compositions.