Problem 58 - Elementary and Intermediate Algebra

Question

Find the midpoint of the line segment with the given endpoints. \((10,4),(2,-2)\)

Step-by-Step Solution

Answer

The midpoint of the line segment is \((6, 1)\).

Step 1: Identify the Coordinates

The coordinates of the endpoints of the line segment are given as \((x_1, y_1) = (10, 4)\) and \((x_2, y_2) = (2, -2)\).

Step 2: Use the Midpoint Formula

The formula to find the midpoint, \(M\), of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\).

Step 3: Calculate the X-Coordinate of the Midpoint

Substitute the values \(x_1 = 10\) and \(x_2 = 2\) into the formula: \( \frac{10 + 2}{2} = \frac{12}{2} = 6 \).

Step 4: Calculate the Y-Coordinate of the Midpoint

Substitute the values \(y_1 = 4\) and \(y_2 = -2\) into the formula: \( \frac{4 + (-2)}{2} = \frac{2}{2} = 1 \).

Step 5: Write the Midpoint Coordinates

Combine the calculated x and y coordinates to write the midpoint as \((6, 1)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe the geometric relationships between points, lines, and shapes on a coordinate plane. This method of geometry allows us to work with precise location and measurement using numerical coordinates.

The coordinate plane itself consists of two perpendicular lines, typically known as the x-axis (horizontal) and the y-axis (vertical). Together, they create a grid system that enables us to specify any point within the plane using a pair of numbers, known as coordinates.

• The first number, or the x-coordinate, indicates how far along the x-axis the point is.
• The second number, or the y-coordinate, shows the position along the y-axis.

Understanding this system is essential when analyzing line segments and finding midpoints, as each point of interest is built upon these foundational coordinates.

Line Segment Endpoints

In coordinate geometry, a line segment is defined by its two endpoints. These endpoints are essentially fixed points on the plane that determine the start and the end of the segment. For instance, if we're given the endpoints \((10,4)\) and \((2,-2)\), each pair represents a point on the plane.

Understanding endpoints is key to working with line segments, as they serve as the reference points for determining all other features associated with the line segment, including its midpoint.Here’s what you need to remember about endpoints:

• They are constant; they do not change unless you redefine the segment.
• Almost all calculations involving the segment, such as length and slope, hinge on these endpoints.

A clear grasp of endpoint coordinates enables you to perform further calculations, like finding the segment's midpoint using the midpoint formula.

Midpoint Calculation

The midpoint formula is a straightforward yet powerful tool in coordinate geometry. It allows you to find the exact center point of a line segment by averaging the x and y coordinates of its endpoints.

**Here's how to use the midpoint formula:**Let’s say we have a line segment with endpoints \((x_1, y_1) = (10, 4)\) and \((x_2, y_2) = (2, -2)\). To find the midpoint, \(M\), we calculate:- **X-coordinate of midpoint**: \( \frac{x_1 + x_2}{2} = \frac{10 + 2}{2} = 6 \)- **Y-coordinate of midpoint**: \( \frac{y_1 + y_2}{2} = \frac{4 + (-2)}{2} = 1 \)This results in the midpoint \((6, 1)\), meaning that the midpoint is exactly half-way along the segment, both in terms of vertical and horizontal space.

This calculation enables you to find balanced or center points effectively, critical for breaking geometric problems into more manageable parts.