Problem 45 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

The midpoint is \((0, 2b)\).

Step 1: Identify the Coordinates

The coordinates of the two points are given as \((a, b)\) and \((-a, 3b)\). Let's assign these coordinates to points \(A\) and \(B\) respectively: \(A(a, b)\) and \(B(-a, 3b)\).

Step 2: Use the Midpoint Formula

The midpoint formula provides the coordinates of the midpoint \((M)\) of the line segment connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) as: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]For this problem, substitute \(x_1 = a\), \(y_1 = b\), \(x_2 = -a\), and \(y_2 = 3b\).

Step 3: Calculate the x-coordinate of the Midpoint

To find the x-coordinate of the midpoint, apply the midpoint formula:\[ x_{mid} = \frac{a + (-a)}{2} = \frac{a - a}{2} = \frac{0}{2} = 0 \]

Step 4: Calculate the y-coordinate of the Midpoint

To find the y-coordinate of the midpoint, apply the midpoint formula:\[ y_{mid} = \frac{b + 3b}{2} = \frac{4b}{2} = 2b \]

Step 5: Write the Midpoint Coordinates

Combine the x and y coordinates to write the coordinates of the midpoint:The midpoint \(M\) is \((0, 2b)\).

Key Concepts

Coordinates

Coordinates are a fundamental part of the way we describe locations on a plane. They provide a unique label for each point, much like an address for a house. Coordinates come in pairs when we're working in a two-dimensional context, such as on a piece of paper or a computer screen. This pair is written as

• x-coordinate, which indicates a position left or right of the origin (zero point)
• y-coordinate, which indicates a position up or down from the origin

Imagine a plane divided by two lines, one horizontal (the x-axis) and one vertical (the y-axis), both meeting at a zero point, known as the origin. Every point's location can be described relative to this origin using coordinates, which we write in the form \( (x, y) \).
In the given exercise, the coordinates of point A are \( (a, b) \), indicating it is located 'a' units along the x-axis and 'b' units along the y-axis. Point B's coordinates are \( (-a, 3b) \), showing it is mirrored to the left by 'a' units and 'up' by thrice distance of 'b' along the y-axis.

Line Segment

A line segment is the part of a line that connects two points. It is distinguished because it has two endpoints, unlike a line that extends infinitely in both directions. In the context of a Cartesian plane, a line segment can be defined using the coordinates of its endpoints.
For example, if the endpoints of a segment are \( A(a, b) \) and \( B(-a, 3b) \), the segment itself represents the shortest path connecting these points.
Understanding the properties of line segments is crucial for various geometric calculations, such as finding midpoints, lengths, or determining collinearity. In the exercise, by calculating the midpoint, you find a point that equally divides the line segment into two smaller segments, each of equal length.

Algebraic Expressions

Algebraic expressions are like a language used to describe mathematical ideas using symbols, numbers, and letters. These expressions represent quantities that can change, often acting as placeholders for specific values.
In the exercise above, letters like \( a \) and \( b \) are used in the coordinates of points as algebraic variables. They allow for generalization, indicating that while the specific values are unknown, they can be calculated or estimated given additional information.
Expressions are the foundation of every calculation in this context. When calculating the midpoint, algebraic methods can simplify the combination of coordinates by treating them as variables in an equation:

• The average of the x-coordinates: \( \frac{a + (-a)}{2} \)
• The average of the y-coordinates: \( \frac{b + 3b}{2} \)

These algebraic expressions offer a straightforward way to compute without numerical values, making them powerful tools in geometry and beyond.