Problem 91 - Intermediate Algebra

Question

Find the midpoint of the line segment whose endpoints are given. See Example 7 $$ (7,3),(-1,-3) $$

Step-by-Step Solution

Answer

The midpoint of the line segment is (3, 0).

Step 1: Understand the Problem

The problem requires us to find the midpoint of a line segment given its endpoints. The formula for the midpoint \((x_m, y_m)\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \(x_m = \frac{x_1 + x_2}{2}\) and \(y_m = \frac{y_1 + y_2}{2}\).

Step 2: Identify the Coordinates

We are given the endpoints of the line segment: Point A is \((7, 3)\) and Point B is \((-1, -3)\). These coordinates will be inserted into the midpoint formula.

Step 3: Calculate the Midpoint X-Coordinate

Apply the formula for the x-coordinate of the midpoint: \(x_m = \frac{x_1 + x_2}{2}\). Substitute \(x_1 = 7\) and \(x_2 = -1\). Calculate \(x_m = \frac{7 + (-1)}{2} = \frac{6}{2} = 3\).

Step 4: Calculate the Midpoint Y-Coordinate

Apply the formula for the y-coordinate of the midpoint: \(y_m = \frac{y_1 + y_2}{2}\). Substitute \(y_1 = 3\) and \(y_2 = -3\). Calculate \(y_m = \frac{3 + (-3)}{2} = \frac{0}{2} = 0\).

Step 5: State the Midpoint

Combine the calculated x and y coordinates to express the midpoint. The midpoint of the line segment is \((3, 0)\).

Key Concepts

Coordinate Geometry

Coordinate Geometry is a field in mathematics where geometry is studied using a coordinate system. This system allows precise descriptions of geometric figures and relationships in terms of numerical coordinates. By using ordered pairs \(x, y\) on a two-dimensional plane, we can perform various calculations, such as finding distances or midpoints.

Understanding coordinate geometry helps in visualizing linear relationships between points on a plane. Some basic operations include identifying specific points, measuring angles and lengths, and analyzing slopes of lines.

• Allows conversion of geometric problems into algebraic equations.
• Makes it easier to handle figures analytically.
• Helps in linking algebraic equations with geometric figures.

Line Segment

A line segment is a part of a line that has two endpoints. Unlike a line that extends infinitely in both directions, a line segment is finite and can be easily identified by its endpoints. In coordinate geometry, a line segment can be described using the coordinates of its endpoints: for example, \((7, 3)\) and \((-1, -3)\).

Knowing these endpoints allows us to perform a variety of calculations, such as determining its length or finding its midpoint. Line segments offer a concise way to handle problems involving finite sections on a plane.

• Represents the shortest path between two points in a plane.
• Essential in calculating midpoints and lengths.
• Can be used to define polygons and other geometric shapes.

Midpoint Calculation

Calculating the midpoint of a line segment in coordinate geometry involves using the midpoint formula. This formula creates a new point that is exactly in the middle of the two endpoints. The formula for finding the midpoint \(x_m, y_m\) between two points \(x_1, y_1\) and \(x_2, y_2\) is given by:

\[x_m = \frac{x_1 + x_2}{2}\],\[y_m = \frac{y_1 + y_2}{2}\].

This formula calculates an average of the x and y coordinates, thereby locating the midpoint. Using the given endpoints \((7, 3)\) and \((-1, -3)\), we plug these into the formula to find the midpoint \(3, 0\).

• Essential for dividing line segments precisely.
• Simplifies understanding of geometric properties.
• Utilizes basic arithmetic operations.

Mathematical Formulas

Mathematical formulas serve as powerful tools in solving various problems related to coordinate geometry. They provide a standardized way to perform calculations such as measuring distance, finding midpoints, and calculating areas. In the context of this exercise, the midpoint formula simplifies finding the central point of a straight line segment.

A good grasp of mathematical formulas allows students to efficiently solve complex problems by converting visual geometric problems into numeric operations. They form the backbone of problem-solving in mathematics.

• Instructive for deriving consistent results.
• Provide shortcuts to lengthy calculations.
• Make complex problems more approachable and less daunting.