Problem 23 - College Algebra

Question

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

Step-by-Step Solution

Answer

The midpoint of the line segment with the given endpoints is \( ( \frac{3}{2},-6)\)

Step 1: Determine the Endpoints

In this case, the endpoints of the line segment are given as \((-3,-4)\) and \( (6,-8)\). So, \(x_1 = -3\), \(y_1 = -4\), \(x_2 = 6\), and \(y_2 = -8\).

Step 2: Apply the Midpoint Formula

Now we apply the midpoint formula \((\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2})\) for the given endpoints. Substitute the values we have into the equation; \((\frac{{(-3) + (6)}} {2}, \frac{{(-4) + (-8)}}{2})\).

Step 3: Calculate the Midpoint

After calculating the fractions we get; \((\frac{3}{2}, \frac{-12}{2})\). Thus, \(( \frac{3}{2},-6)\) is the midpoint of the line segment with the given endpoints.

Key Concepts

Endpoints

Endpoints are the specific coordinates that mark the extremities or ends of a line segment in coordinate geometry. Each endpoint is defined by a pair of coordinates \(x, y\). In the exercise, the endpoints are given as \((-3,-4)\) and \(6,-8)\). Here, \(x_1 = -3\) and \(y_1 = -4\) pertain to the first endpoint, while \(x_2 = 6\) and \(y_2 = -8\) apply to the second one.

Understanding endpoints is crucial because they provide the necessary points to perform various calculations, such as finding the midpoint or the length of a line segment. When given endpoints:

• Always identify each coordinate accurately to avoid mistakes in calculations.
• Remember that swapping coordinates can lead to errors, especially when applying formulas.

Endpoints are not just any arbitrary numbers; they represent precise locations on a graph that are essential for solving problems in coordinate geometry.

Line Segment

A line segment is a part of a line that is bounded by two distinct endpoints. Unlike a line, it does not extend infinitely in both directions. Instead, it is finite and measured only between its two endpoints.

For example, in our exercise, the line segment is defined by the endpoints \((-3, -4)\) and \(6, -8)\). This means the line segment is only the section of the line lying between these two points. Finding the midpoint or calculating the length of a line segment in coordinate geometry requires knowledge of these endpoints.

Key properties of a line segment include:

• It is the shortest path connecting two points.
• Each of its points can be calculated or found using linear interpolation between the endpoints.
• It aids in visualizing the geometric properties and relationships in problems involving multiple points or figures.

Thus, understanding what a line segment is helps in effectively solving problems involving distances, midpoints, and equations of lines in coordinate geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry, is a branch of geometry where the positions of points on a plane are described using an ordered pair of numbers. This allows for the exploration and solution of geometric problems using algebraic equations and principles.

In solving the exercise problem, we use coordinate geometry to find the midpoint of a line segment defined by two points. The coordinates form the basis for many standard formulas, such as the midpoint formula and the distance formula. In coordinate geometry:

• Each point is represented as \((x, y)\) on a 2-dimensional plane.
• Geometric interpretations aid in visualizing algebraic relationships between points and figures.
• Formulas like \(\frac{(x_1 + x_2)}{2}, \frac{(y_1 + y_2)}{2}\) for the midpoint simplify solving complex geometric problems.

Coordinate geometry provides a precise language and framework for analyzing and solving geometric and spatial problems, making it an indispensable tool in mathematics.