Problem 4 - Algebra 2

Question

Find the midpoint of the line segment with endpoints at the given coordinates. $$ (-12,-2),(-3.5,-7) $$

Step-by-Step Solution

Answer

The midpoint is \((-7.75, -4.5)\).

Step 1: Understanding the Midpoint Formula

The midpoint of a line segment is found using the formula \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the endpoints.

Step 2: Identifying the Endpoint Coordinates

The given endpoints are \((-12, -2)\) and \((-3.5, -7)\). Therefore, \(x_1 = -12\), \(y_1 = -2\), \(x_2 = -3.5\), and \(y_2 = -7\).

Step 3: Substituting Coordinates into the Midpoint Formula

Substitute the coordinates into the midpoint formula: \[ M = \left( \frac{-12 + (-3.5)}{2}, \frac{-2 + (-7)}{2} \right) \]

Step 4: Calculating the Midpoint

Calculate each component of the midpoint: First, for the x-coordinate: \[ \frac{-12 + (-3.5)}{2} = \frac{-15.5}{2} = -7.75 \] Next, for the y-coordinate: \[ \frac{-2 + (-7)}{2} = \frac{-9}{2} = -4.5 \] Thus, the midpoint is \((-7.75, -4.5)\).

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that deals with representing geometric shapes and figures using coordinate systems. This system allows us to easily locate points, lines, and other figures in a plane using ordered pairs, known as coordinates. In a two-dimensional space, any point can be described with two numbers, \(x\) and \(y\), which represent the horizontal and vertical positions of the point, respectively.
Coordinate geometry is valuable because it provides a way to solve geometric problems using algebraic techniques. You can translate many geometric problems, like finding distances or midpoint, into algebraic equations.

• Every point in this system is represented by \( (x, y) \).
• It helps in finding the distances, angles, and midpoints between points.

The beauty of coordinate geometry lies in its ability to provide a clear visual representation, which helps simplify mathematical concepts and calculations.

Line Segments

A line segment is a part of a line that is bounded by two distinct endpoints. In coordinate geometry, we often refer to segments by their endpoints, like \( (-12, -2) \) and \( (-3.5, -7) \). Unlike a line, which extends forever in both directions, a line segment has a finite length. Understanding the properties of line segments helps in calculating aspects like length and midpoint.
In our original exercise, the focus is on finding the midpoint of such a segment. The midpoint creates two equal halves of the segment.

• Endpoints are necessary to define a line segment.
• The midpoint divides the segment into two equal parts.

Real-world examples include the sides of a rectangle or the distance between two fixed points on a map, both of which can be analyzed as line segments in a coordinate plane.

Algebraic Calculation

Algebraic calculation involves using algebraic methods to solve mathematical problems. In the context of our exercise, it refers to applying the midpoint formula to find a precise point value. The midpoint formula itself is derived by averaging the x-coordinates and y-coordinates of the endpoints.
Using the given endpoints, substitute into the formula:

• \[ M = \left( \frac{-12 + (-3.5)}{2}, \frac{-2 + (-7)}{2} \right) \]
• Calculate: \( \frac{-15.5}{2} = -7.75 \) and \( \frac{-9}{2} = -4.5 \)

This results in the midpoint being located at \( (-7.75, -4.5) \). Algebraic calculation is a systematic method that ensures accurate results. It's a crucial part of solving real-life problems that can be modeled by such mathematical exercises. By breaking down problems into step-by-step operations, we simplify complex calculations to reach intended results.