Problem 24 - College Algebra

Question

Find the midpoint of each line segment with the given endpoints. $$(-2,-1)\( and \)(-8,6)$$

Step-by-Step Solution

Answer

The midpoint of the line segment with endpoints \((-2,-1)\) and \((-8,6)\) is \((-5, 2.5)\)

Step 1: Identify the coordinates of the endpoints

Recognize the given points as \((-2, -1)\) and \((-8, 6)\). These correspond to \((x_1, y_1)\) and \((x_2, y_2)\) respectively in the midpoint formula.

Step 2: Apply the Midpoint formula

Plugging the coordinates we identified in step 1 into the midpoint formula, we can find the midpoint of the line segment. The x-coordinate of the midpoint is \((x_1 + x_2)/2 = (-2 - 8)/2 = -10/2 = -5\). Similarly, the y-coordinate of the midpoint is \((y_1 + y_2)/2 = (-1 + 6)/2 = 5/2 = 2.5\).

Step 3: Write the Midpoint

Combine the x and y coordinates from step 2 to form the coordinates of the midpoint, giving us \((-5, 2.5)\).

Key Concepts

Understanding Line Segments

A line segment is a part of a line that has two endpoints. Unlike a line, which stretches infinitely in both directions, a line segment has a start and an end. In our exercise, the endpoints are given as coordinates, specifically

• \((-2, -1)\)
• \((-8, 6)\)

These points define the boundary of the line segment. You can think of a line segment as a "cut piece" of a line. It’s like taking a string and cutting it to a specific length. This concept is crucial in geometry and helps us navigate the notion of distance and position on a plane. Understanding a line segment helps to find midpoints, distances, and other geometric properties.

What Coordinates Communicate

Coordinates are pairs of numbers used to determine the position of a point on a plane. There are two values: the first is the x-coordinate, and the second is the y-coordinate. They are usually written in parentheses, like

• \((x, y)\)

Coordinates are like addresses for points in the plane. They inform us precisely where a point lies in relation to the two reference axes, the x-axis, and the y-axis. In our example, the points \((-2, -1)\) and \((-8, 6)\) tell us the exact location of the endpoints of our line segment on a Cartesian plane. With this information, we can easily locate and plot the points, helping us in calculations like finding the midpoint.

Exploring the X-Coordinate

The x-coordinate is the first number in an ordered pair and tells us how far left or right a point is along the x-axis. It represents the horizontal position. A positive x-value indicates the point is to the right of the origin, while a negative value means it's to the left. For our points:

• \((-2, -1)\) - here the x-coordinate is \(-2\). This tells us the point is two units to the left on the x-axis.
• \((-8, 6)\) - with an x-coordinate of \(-8\), indicating it's eight units left of the origin.

When we compute the midpoint, we average these x-coordinates to find a middle position horizontally between the two endpoints. This balancing of positions is essential for accurately locating the midpoint.

Grasping the Y-Coordinate

The y-coordinate is the second value in an ordered pair, indicating how far up or down a point is on the y-axis. It's all about vertical placement. Positive y-values mean the point is above the x-axis, and negative values show it's below. In the current problem:

• \((-2, -1)\) - here, the y-coordinate is \(-1\), placing it one unit below the x-axis.
• \((-8, 6)\) - has a y-coordinate of \(6\), positioning it six units above the x-axis.

Calculating the midpoint involves finding the average of these y-values, which indicates the central vertical position between the two endpoints. This ensures that our midpoint truly sits at the middle in terms of vertical distance as well.