Problem 46 - College Algebra with Modeling and Visualization

Question

Find the midpoint of the line segment connecting the points. $$ (-a, b),(3 a, b) $$

Step-by-Step Solution

Answer

The midpoint is \((a, b)\).

Step 1: Plot the Problem

We need to find the midpoint of the line segment connecting the points \((-a, b)\) and \((3a, b)\). The coordinates are given in the form \((x_1, y_1)\) and \((x_2, y_2)\) respectively.

Step 2: Apply Midpoint Formula

The formula to find the midpoint \(M\) of a line segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\).

Step 3: Substitute Values

Substituting the given values \((-a, b)\) for \((x_1, y_1)\) and \((3a, b)\) for \((x_2, y_2)\) into the formula, we have:\[M = \left(\frac{-a + 3a}{2}, \frac{b + b}{2}\right)\]

Step 4: Simplify the Expression

Compute each component of the midpoint:- For the \(x\) component: \( \frac{-a + 3a}{2} = \frac{2a}{2} = a \)- For the \(y\) component: \( \frac{b + b}{2} = \frac{2b}{2} = b \)Thus, the midpoint is \((a, b)\).

Key Concepts

Understanding Line Segments

A line segment is a part of a line that is bounded by two distinct endpoints and contains every point on the line between its endpoints. It is like a slice of a whole line that has a start and an end. Unlike a line, which extends infinitely in both directions, a line segment is finite.In coordinate geometry, we usually identify line segments by their endpoints, which are given in the form of coordinates. For example, in our problem, the points \((-a, b)\) and \((3a, b)\) are the endpoints of our line segment.Understanding these endpoints is crucial because the midpoint of a line segment is always somewhere on the segment itself, equidistant from both endpoints. This makes the concept of the midpoint particularly useful in geometry. It helps in simplifying problems that involve bisecting line segments or finding centers of symmetrical shapes.

Exploring Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses a coordinate plane to represent and solve geometric problems. It combines algebra and geometry by using coordinates to describe geometric figures like lines and curves.Key concepts to understand include:

• Coordinates: Ordered pairs \((x, y)\) that indicate a precise location on the plane.
• Axes: The horizontal (x-axis) and vertical (y-axis) lines that intersect at the origin (0, 0).
• Origin: The point where both axes meet, at \((0, 0)\).

In problems involving line segments, we utilize coordinates to compute various properties, such as length, slope, and midpoint. When these points are plotted on a graph, they offer a visual interpretation of the relationships between points and lines, making it easier to solve problems algebraically.

Working with Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They help us represent general mathematical relationships in a compact and symbolic form.In the context of midpoint calculations:

• Variables are often used to symbolically represent coordinates, like \(a\) in the points \((-a, b)\) and \((3a, b)\). This allows for general solutions that can be adapted to specific numbers later on.
• Operations such as addition and division are used to manipulate these coordinates according to formulas, like finding a midpoint using \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\).
• Simplifying expressions is crucial to get final answers. For example, simplifying \(\frac{-a + 3a}{2}\) to \(a\) demonstrates how algebra makes the problem solvable.

By harnessing algebraic expressions, we can translate the geometric properties into a language of math, making it easier to calculate outcomes like midpoints, distances, and intersections.