Problem 24 - College Algebra

Question

For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points. $$(-5,-3)\text { and }(-2,-8)$$

Step-by-Step Solution

Answer

The midpoint is \((-3.5, -5.5)\).

Step 1: Understanding the Midpoint Formula

The formula to find the midpoint of a line segment given two endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). We'll use this formula to find the midpoint of the line segment joining the points \((-5, -3)\) and \((-2, -8)\).

Step 2: Applying the Midpoint Formula

Plug the given points \((-5, -3)\) and \((-2, -8)\) into the midpoint formula. For the x-coordinates: \(\frac{-5 + (-2)}{2}\). For the y-coordinates: \(\frac{-3 + (-8)}{2}\).

Step 3: Calculating the x-coordinate

Calculate the x-coordinate of the midpoint: \(-5 + (-2) = -7\). Therefore, the x-coordinate of the midpoint is \(\frac{-7}{2} = -3.5\).

Step 4: Calculating the y-coordinate

Calculate the y-coordinate of the midpoint: \(-3 + (-8) = -11\). Therefore, the y-coordinate of the midpoint is \(\frac{-11}{2} = -5.5\).

Step 5: Conclusion

The coordinates of the midpoint of the line segment joining the points \((-5, -3)\) and \((-2, -8)\) are \((-3.5, -5.5)\).

Key Concepts

Coordinates

Coordinates are essential in geometry as they help us locate points on a plane. Essentially, they are like addresses or labels for points. Each point on a plane has a pair of coordinates, denoted as \(x, y\).

• The first number is the x-coordinate, which tells us how far the point is from the vertical or y-axis.
• The second number is the y-coordinate, which tells us how far the point is from the horizontal or x-axis.

In our exercise, we work with two points: \((-5, -3)\) and \((-2, -8)\). These coordinates tell us the exact position of each endpoint of the line segment in question. Understanding how coordinates work is crucial for calculating midpoints, distances, and other geometric properties.

Line Segment

A line segment is a fundamental part of geometry, consisting of two endpoints and all the points in between them. Unlike a line, a line segment does not extend infinitely.

• It has a finite length, measured directly between its two endpoints.
• It is represented with a straight path connecting the endpoints.

In our exercise, the line segment connects the points \((-5, -3)\) and \((-2, -8)\). Our task is to find the midpoint of this line segment. The midpoint essentially divides the segment into two equal halves, giving us an average location between the two endpoints.

Distance Formula

The distance formula is a technique used in geometry to find the distance between two points on a plane. Although it is not directly used to find the midpoint, understanding the concept is beneficial.

• The formula for finding the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.\]
• It is derived from the Pythagorean theorem and helps us understand how far apart two points are.

For our specific points, \((-5, -3)\) and \((-2, -8)\), knowing the distance can give insights into the geometry of the line segment, although finding the midpoint doesn't require the distance formula directly.

Geometry

Geometry is the branch of mathematics dealing with shapes, sizes, and the properties of space. It's applicable in numerous everyday contexts, including the spatial arrangement of objects and the movement of paths.

• Line segments, midpoints, and distances are all core topics within geometry.
• Geometry helps us visualize and solve problems involving spatial relationships.

In dealing with the line segment joining points \((-5, -3)\) and \((-2, -8)\), we applied the geometric concept of finding a midpoint to solve the exercise. Midpoints play a crucial role in geometric constructions, allowing us to divide segments neatly and symmetrically into two equal parts.