Problem 21 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Endpoint Coordinates

The coordinates of the endpoints are identified as (-2,-8) and (-6,-2), so \(x_1 = -2\), \(y_1 = -8\), \(x_2 = -6\), and \(y_2 = -2\).

Step 2: Apply Midpoint Formula to x-coordinates

To find the x-coordinate of the midpoint, use the formula \(x = (x_1 + x_2)/2\). Substituting the known values, \(x = (-2 + -6)/2 = -4\).

Step 3: Apply Midpoint Formula to y-coordinates

To find the y-coordinate of the midpoint, use the formula \(y = (y_1 + y_2)/2\). Substituting the known values, \(y = (-8 + -2)/2 = -5\).

Step 4: Write the Midpoint Coordinates

The coordinates of the midpoint will thus be the calculated x and y values arranged as a pair, which results in the midpoint is (-4, -5).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry that uses a coordinate system to investigate geometrical shapes and lines. This blend of algebra and geometry allows for the precise analysis of points, lines, and shapes in the two-dimensional plane.

In the case of finding the midpoint of a line segment, coordinate geometry simplifies the problem by translating it into an algebraic equation. Coordinates are an ordered pair of numbers which define the position of points in a two-dimensional space, usually referred to with the variables x (horizontal axis) and y (vertical axis). By using these coordinates, we can easily visualize and solve geometric problems such as finding midpoints, calculating distances, and determining slope.

Algebraic Methods

Algebraic methods help us to solve problems involving numbers and variables, making them ideal for dealing with geometric issues. In the midpoint exercise, we see algebra at work when we apply the midpoint formula.

The midpoint formula is \( (x_1 + x_2)/2, (y_1 + y_2)/2 \), where \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of the endpoints of a line segment. This formula mathematically computes the arithmetic mean, or average, of the endpoints' x and y coordinates separately, giving the precise center or 'midpoint' of the line.

Why is this algebraic?

Because we use the arithmetic operations of addition and division, and we work with variables to denote unknown or general values.

Geometry Problem Solving

Geometry problem solving is the process of determining geometric quantities and understanding the properties of shapes and spaces, which often involves a series of logical steps.

When tackling a problem like finding the midpoint, a structured approach is necessary:

• First, identify and plot the given coordinates.
• Then, apply an appropriate mathematical formula—in this case, the midpoint formula.
• Finally, calculate and interpret your results within the context of the geometric problem.

This systematized approach not only leads to the solution but also brings clarity and understanding to the principles and concepts involved. In the solution given, each step layers on the next, combining mathematical formulas with geometric understanding to accomplish the task at hand.