Problem 21 - College Algebra

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 given line segment is (-4, -5).

Step 1: Identify the coordinates of the endpoints

The coordinates of the first point, A, are (-2,-8). The coordinates of the second point, B, are (-6,-2). So, x1 = -2, y1 = -8, x2 = -6 and y2 = -2.

Step 2: Apply the midpoint formula

The midpoint M(x, y) of the line segment with endpoints A(x1, y1) and B(x2, y2) is given by \(x = \frac{x1 + x2}{2}\) and \(y = \frac{y1 + y2}{2}\). So, applying the midpoint formula, we get to calculate x-coordinate : \(x = \frac{-2 + (-6)}{2} = -4 \), and the y-coordinate : \(y = \frac{-8 + (-2)}{2}= -5\).

Step 3: Present the final answer

Therefore, the coordinates of the midpoint M(x, y) of the given line segment with endpoints A(-2,-8) and B(-6,-2) is (-4, -5).

Key Concepts

Understanding Coordinates

Coordinates are essential to understanding how we locate points on a plane. A coordinate consists of two numbers, written in the form

• the first number (x-coordinate) indicates the position of the point along the horizontal axis (x-axis), and
• the second number (y-coordinate) shows the position along the vertical axis (y-axis).

Each point in a 2D space symbolizes a unique set of coordinates. In our problem, the endpoints of the line segment are given as (-2,-8) and (-6,-2).
Here, (-2,-8) means the point is located 2 units to the left and 8 units below the origin. Similarly, (-6,-2) is 6 units to the left and 2 units below the origin. Understanding how these coordinates define a position in space is crucial when calculating distances, midpoints, or other attributes related to line segments.

Exploring the Concept of a Line Segment

A line segment is something that represents a part of a line. It has two endpoints and includes all the points between them. Unlike a line that extends indefinitely in both directions, a line segment has a definite start and end point.
In our particular example, the line segment is drawn from point A (-2,-8) to point B (-6,-2). This means our line segment will stay within these two boundaries.
It's helpful to remember:

• A line segment connects two points.
• This segment will include each point directly between these two endpoints.
• Understanding line segments helps in determining things like length and midpoint calculation.

Since line segments are so foundational in geometry, grasping their basics makes it easier to tackle complex geometric problems.

Endpoint Calculation and the Midpoint Formula

When you have two endpoints of a line segment, like in our problem, you can calculate many different characteristics. One important calculation is finding the midpoint of this segment. The midpoint formula helps us find the exact center point between two endpoints.
The midpoint formula is:

• For the x-coordinate: \( \frac{x1 + x2}{2} \)
• For the y-coordinate: \( \frac{y1 + y2}{2} \)

The midpoint is the point that divides the segment into two equal halves. Applying this to our endpoints:

• Calculate the midpoint's x-coordinate: \( \frac{-2 + (-6)}{2} = -4 \)
• Calculate the midpoint's y-coordinate: \( \frac{-8 + (-2)}{2} = -5 \)

So, the midpoint of our line segment is (-4, -5).
This midpoint is vital in geometry for dividing line segments and finding centroids in more complex figures.