Problem 121 - Elementary and Intermediate Algebra

Question

Consider the line that passes through \(P(-2,3)\) and \(Q(4,-4)\). Find the midpoint of line segment \(P Q\).

Step-by-Step Solution

Answer

The midpoint of line segment PQ is \\( (1, -\frac{1}{2}) \\).

Step 1: Understand the Midpoint Formula

The midpoint of a line segment connecting two points \(x_1, y_1\) and \(x_2, y_2\) can be found using the formula \( ext{Midpoint} = rac{(x_1 + x_2)}{2}, rac{(y_1 + y_2)}{2} \). This formula calculates the average of the x-coordinates and the y-coordinates to find the point exactly halfway between the two points.

Step 2: Identify Coordinates of Points

The coordinates of the given points are \(P(-2, 3)\) and \(Q(4, -4)\). So, we identify \(x_1 = -2\), \(y_1 = 3\), \(x_2 = 4\), and \(y_2 = -4\).

Step 3: Substitute the Values into the Midpoint Formula

Substitute \(x_1 = -2\), \(x_2 = 4\), \(y_1 = 3\), and \(y_2 = -4\) into the midpoint formula. Calculate \( rac{(-2 + 4)}{2}, rac{(3 + (-4))}{2} \).

Step 4: Calculate the Midpoint

The x-coordinate of the midpoint is \( rac{(-2 + 4)}{2} = rac{2}{2} = 1 \). The y-coordinate of the midpoint is \( rac{(3 + (-4))}{2} = rac{-1}{2} = -rac{1}{2} \). Thus, the midpoint is \( (1, -rac{1}{2}) \).

Key Concepts

Understanding Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where we use coordinates and algebra to study and define geometric structures. This approach allows us to calculate distances, find midpoints, and analyze the properties of various shapes. By using a coordinate plane, which consists of a horizontal axis (x-axis) and a vertical axis (y-axis), each point can be uniquely identified by a pair of numbers known as coordinates.
The basic idea is to translate geometric problems into algebraic ones, allowing them to be solved using equations and functions. For example, by examining how points, lines, and shapes are represented in this coordinate system, we can derive formulas that help solve complex problems, such as finding the midpoint of a line segment.

Breaking Down a Line Segment

A line segment is part of a line that is bounded by two distinct endpoints and contains every point on the line between its endpoints. In the context of coordinate geometry, a line segment can be represented on a coordinate plane by its endpoints, which are described by coordinate pairs.
When you have two points in a coordinate system, such as point \( P(-2,3) \) and point \( Q(4,-4) \), the line segment \( PQ \) directly connects them. This linear connection allows us to use the properties of lines in geometry to analyze further. Calculating measures such as length, slope, and midpoint of this line segment helps understand the segment's geometric properties.

• The length of a line segment can be found using the distance formula.
• The midpoint helps in finding the exact middle point between two endpoints.

Understanding the characteristics of a line segment builds foundational skills in solving many geometric problems.

Calculating Midpoints with Average Coordinates

The concept of average coordinates is key to determining the midpoint of a line segment. By finding the arithmetic mean, or average, of the x-coordinates and y-coordinates of the endpoints, we can easily locate the midpoint: the point that lies exactly halfway between them.
To use the midpoint formula:
Given two endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \), the formula for the midpoint \( M \) is: \[ M = \left( \frac{(x_1 + x_2)}{2}, \frac{(y_1 + y_2)}{2} \right) \]
This equation calculates the average of the coordinates. The result gives you a new coordinate pair that represents the midpoint.

• This method is straightforward and reliable for finding midpoints.
• Helps in tasks ranging from simple geometry problems to complex data algorithms.

Applying this formula to the endpoints of \( P(-2, 3) \) and \( Q(4, -4) \) in our example gives a midpoint of \( (1, -\frac{1}{2}) \), which is the calculated center of the line segment.