Problem 21 - College Algebra Essentials

Question

find the midpoint of each line segment with the given endpoints. $$ (-2,-8) \text { and }(-6,-2) $$

Step-by-Step Solution

Answer

The midpoint of the line with the endpoints (-2,-8) and (-6,-2) is (-4,-5).

Step 1: Identify the coordinates

Identify the points given and label them. The first endpoint is (-2,-8) so \(x1=-2\) and \(y1=-8\). Similarly, the second endpoint is (-6,-2), so \(x2=-6\) and \(y2=-2\).

Step 2: Substitute into the formula

Substitute the values of \(x1, y1, x2,\) and \(y2\) into the formula \((\frac{x1+x2}{2} , \frac{y1+y2}{2})\). This means calculate \((\frac{-2+(-6)}{2} , \frac{-8+(-2)}{2})\) which is \((\frac{-8}{2} , \frac{-10}{2})\) .

Step 3: Solve the expression

Solve the expression to find the numerical answer. The result is \((-4, -5)\).

Key Concepts

Midpoint Formula

The midpoint of a line segment is the point that divides the segment into two equal parts. It is located exactly halfway between the two endpoints of the segment. To find the midpoint in a coordinate plane, you use the midpoint formula, which is expressed as \( (\frac{x1 + x2}{2} , \frac{y1 + y2}{2}) \).

This formula computes the average of the x-coordinates and y-coordinates of the endpoints, thus locating the center point. For example, given the endpoints (-2, -8) and (-6, -2), plug the values into the midpoint formula like this: \( (\frac{-2 + (-6)}{2}, \frac{-8 + (-2)}{2}) \). Simplifying the fractions gives you the coordinates of the midpoint, which in this case is (-4, -5).

Understanding this formula is crucial for various applications in geometry, including constructing bisectors, determining the center of shapes, and even in advanced topics like calculus.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This system allows us to represent geometrical shapes in an algebraic form and solve geometrical problems using algebraic methods.

Points on a plane are defined by two coordinates: the x-coordinate indicates the horizontal position, and the y-coordinate represents the vertical position. When we talk about finding the midpoint, we are working within this coordinate system, applying algebraic formulas to geometric figures.

With the endpoints of a line segment given as (-2, -8) and (-6, -2), coordinate geometry allows us to graph these points, visualize the segment, and calculate its midpoint, thereby connecting algebraic concepts with visual representation and understanding.

Algebraic Methods

Algebraic methods refer to the steps and procedures used to solve equations and manipulate algebraic expressions. These methods are fundamental in simplifying and finding solutions to various problems in mathematics, including those in geometry.

When finding the midpoint of a line segment, we use algebraic manipulation to work with the coordinates of the endpoints. The process involves adding and dividing numbers, which are basic algebraic operations.

For instance, in our example: Firstly, we add the x-coordinates (-2 and -6) and the y-coordinates (-8 and -2). Then, we divide these sums by 2 to apply the midpoint formula. This is a demonstration of how algebra simplifies geometry by turning shapes and figures into solvable equations.