Problem 87 - Algebra A Combined Function

Question

Find the midpoint of each line segment whose endpoints are given. (6,-8)\(;(2,4)\)

Step-by-Step Solution

Answer

The midpoint is (4, -2).

Step 1: Understanding the Midpoint Formula

To find the midpoint of a line segment, we use the formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \), where \((x_1, y_1)\) and \((x_2, y_2)\) are the endpoints of the segment.

Step 2: Substituting the Endpoint Values

Substitute \((x_1, y_1) = (6, -8)\) and \((x_2, y_2) = (2, 4)\) into the midpoint formula. It becomes: \( M = \left( \frac{6 + 2}{2}, \frac{-8 + 4}{2} \right) \).

Step 3: Calculating the Midpoint Coordinates

Calculate each part of the formula: \( \frac{6 + 2}{2} = 4 \) and \( \frac{-8 + 4}{2} = -2 \). Therefore, the midpoint is \( M = (4, -2) \).

Key Concepts

Line Segment

A line segment is a fundamental concept in geometry. It consists of two endpoints and all the points in between them. Think of it as the shortest path connecting these two endpoints. Unlike a line, which extends infinitely in both directions, a line segment has a fixed length.
Visualize the endpoints as two dots on a piece of paper, and the segment is the straight line joining them.

• In this exercise, our endpoints are \( (6, -8) \) and \( (2, 4) \).
• The job is to find the midpoint, which is the point exactly halfway along this segment.

The midpoint divides the line segment into two equal lengths. This is incredibly useful in geometry and aids in dividing areas, designing structures, and solving various mathematical problems.

Coordinate Geometry

Coordinate geometry, also called analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics allows us to use coordinates, such as \( (x, y) \) in two dimensions, to describe the place and position of points, curves, and figures.
Every point in space is represented by a set of numbers, and this gives us a precise method of quantifying geometry. In this context:

• We are using 2-dimensional coordinates: the points \( (6, -8) \) and \((2, 4)\).
• Typically, the x-coordinate determines horizontal movement, and the y-coordinate determines vertical movement.

With coordinate geometry, geometric principles can be expressed and solved. For example, calculating the midpoint of a segment involves manipulating the coordinates of the endpoints using algebraic formulas.

Algebra Concepts

Algebra is a branch of mathematics dealing with symbols and rules for manipulating those symbols. In the problem of finding the midpoint, algebra helps simplify the process. We use the midpoint formula:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

• Algebra simplifies arithmetic operations with numbers—here, with the coordinates of points.
• For points \( (6, -8) \) and \((2, 4)\), the algebraic step is simple and straightforward once the values are substituted into the formula.

The core idea is to add each coordinate pair and divide by two to get the averages. This yields the coordinates of the midpoint, \( M = (4, -2) \), by calculating the mathematical averages of the x and y values independently.