Problem 23 - College Algebra Essentials

Question

find the midpoint of each line segment with the given endpoints. $$(-3,-4) \text { and }(6,-8)$$

Step-by-Step Solution

Answer

The coordinates of the midpoint are (1.5, -6)

Step 1: Identifying the coordinates

The given coordinates are (-3,-4) and (6,-8). Thus, x1 = -3, y1 = -4, x2 = 6, and y2 = -8.

Step 2: Substitute into the formula

Now substitute the values of x1, y1, x2, and y2 into the formula so as to get the coordinates of the midpoint: \(((-3+6)/2, (-4+-8)/2)\).

Step 3: Simplifying

After simplifying, it gives \((3/2, -12/2) = (1.5, -6)\) which is the midpoint of the given line segment.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytical geometry, allows us to study geometric figures through algebraic equations. It uses the coordinate plane, a tool that helps in visualizing points, lines, and shapes in a two-dimensional space. This field of mathematics connects algebra and geometry, providing a way to represent geometric constructs analytically.

When solving problems like finding midpoints of line segments, coordinate geometry is a powerful tool. It enables us to represent points using ordered pairs, such as \((x, y)\), where \(x\) represents the horizontal position and \(y\) represents the vertical position.

To find the midpoint of a line segment, you only need to average the \(x\)-coordinates and \(y\)-coordinates of the endpoints, which are typically given as ordered pairs. Each coordinate pair is managed through a simple formula, making it extremely useful and effective in calculations involving the coordinate plane.

Line Segment

A line segment is part of a line that has two distinct endpoints. Unlike a line that extends infinitely in both directions, a line segment has a definite starting and ending point.

In coordinate geometry, a line segment can be represented using two endpoints, each defined by coordinates on the coordinate plane. For example, with endpoints \((-3, -4)\) and \((6, -8)\), the line segment is the shortest path connecting these two points.

Calculating the midpoint of a line segment is a common problem. It represents the point that is exactly in the middle of the segment. You can find it by applying the midpoint formula:

• Midpoint formula: \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)

By calculating the midpoint, you discover the exact center of the line segment in the coordinate plane, offering insights into the geometric figure's structure.

Coordinate Plane

The coordinate plane, also referred to as the Cartesian plane, consists of a two-dimensional space where points are defined by a pair of numerical coordinates. These coordinates are determined by their horizontal, \(x\), and vertical, \(y\), positions relative to two perpendicular axes that intersect at the origin (0,0).

The plane is divided into four quadrants, allowing every point to be located precisely based on its coordinate pair. This design is essential for graphing functions, shapes, and for performing various calculations, such as the one involving finding the midpoint of a line segment.

In problems requiring the use of the coordinate plane, such as with endpoints at \((-3, -4)\) and \((6, -8)\), the plane assists in accurately placing these points, visualizing the line segment, and subsequently determining the midpoint. Understanding how to navigate the coordinate plane is crucial for anyone working on problems related to coordinate geometry. It provides the foundation for studying more complicated geometric and algebraic concepts.