Problem 19 - College Algebra

Question

Find the midpoint of each line segment with the given endpoints. $$(6,8)\( and \)(2,4)$$

Step-by-Step Solution

Answer

The midpoint of the segment between the points (6,8) and (2,4) is (4, 6).

Step 1: Substitute the endpoints into the midpoint formula

First, substitute the given coordinates into the midpoint formula. So, \( M = \left( \frac{{6 + 2}}{2}, \frac{{8 + 4}}{2} \right) \).

Step 2: Simplify the expressions

Next, perform the arithmetic inside the parentheses. \( M = \left( \frac{8}{2}, \frac{12}{2} \right) = (4, 6) \).

Step 3: Write down the final answer

After calculating, we get the Midpoint, M = (4, 6).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics where we use algebraic equations to describe geometric figures. This discipline allows us to understand shapes and their properties more deeply. Here, each point is described using coordinates on a plane, like a graph in your notebook.

• The **x-coordinate** tells us how far along the horizontal axis the point is.
• The **y-coordinate** tells us how far along the vertical axis the point is.

This powerful tool helps in solving problems related to distance, midpoints, and other properties of geometric figures.

Line Segments

A line segment is part of a line described by two endpoints, and it includes everything in between. Line segments are straightforward and easy to understand:

• They have a fixed length because they have specific starting and ending points.
• In our example, the line segment connects the points (6,8) and (2,4).

Using line segments, you can solve numerous problems in geometry, like finding the midpoint or distance between two points using specific formulas.

Arithmetic Calculations

Arithmetic calculations involve basic mathematical operations such as addition, subtraction, multiplication, and division. In the midpoint formula, these operations are key to finding the precise middle of a segment. Let's look at how it works:

• Add the x-coordinates together, then divide by 2: \( \frac{6 + 2}{2} \)
• Add the y-coordinates together, then divide by 2: \( \frac{8 + 4}{2} \)

These steps ensure that the resulting coordinates represent the exact midpoint of the given line segment on the graph.

Endpoints

Endpoints are the two defined points at the ends of a line segment. They are crucial to many geometric calculations, including the midpoint. In the problem, the endpoints are \((6,8)\) and \((2,4)\):

• These endpoints frame the line segment you’re dealing with.
• Knowing these coordinates allows us to plug into the midpoint formula directly.
• The formula helps calculate the midpoint as a precise value, combining both x and y coordinates.

Handling endpoints effectively can solve or simplify many geometric challenges, offering a clear understanding of the shapes and patterns on a coordinate plane.