Problem 16 - Precalculus: Functions and Graphs

Question

The given numbers are coordinates of points \(A, B,\) and \(C,\) respectively, on a coordinate line. Find the distance. (a) \(d(A, B)\) (b) \(d(B, C)\) (c) \(d(C, B)\) (d) \(d(A, C)\) $$-6,-2,4$$

Step-by-Step Solution

Answer

(a) 4, (b) 6, (c) 6, (d) 10

Step 1: Understand the Problem

You are given three points on a line: -6 (point A), -2 (point B), and 4 (point C). The task is to find the distances between these points.

Step 2: Distance Formula on a Line

The distance between two points on a coordinate line is the absolute value of the difference between the coordinates of those points.

Step 3: Calculate d(A, B)

The distance between points A and B is calculated as follows:\[ d(A, B) = |-6 - (-2)| = |-6 + 2| = |-4| = 4 \]

Step 4: Calculate d(B, C)

The distance between points B and C is calculated as follows:\[ d(B, C) = |-2 - 4| = |-6| = 6 \]

Step 5: Calculate d(C, B)

The distance between points C and B is the same but starting from C, which gives:\[ d(C, B) = |4 - (-2)| = |4 + 2| = |6| = 6 \]

Step 6: Calculate d(A, C)

The distance between points A and C is calculated as follows:\[ d(A, C) = |-6 - 4| = |-10| = 10 \]

Key Concepts

Coordinate Geometry

Coordinate Geometry is an area of mathematics that uses a system of coordinates to represent and analyze the position of points in space. In one-dimensional space, which is simply a line, we use a single coordinate to specify the location of a point. This coordinate is usually represented as a single number, indicating the position relative to a reference point, often called the origin.

The main idea is to use these coordinates to determine distances and relationships between points. In our example, the points are given as numbers -6, -2, and 4, representing points A, B, and C on a number line. This visualization helps us easily calculate distances using arithmetic operations. Coordinate Geometry makes these tasks straightforward, offering a powerful tool to visualize and solve geometric problems in multiple dimensions.

Absolute Value

The concept of Absolute Value is essential when dealing with distances in Coordinate Geometry. Absolute value refers to the non-negative value of a number without regard to its sign, represented by vertical bars, such as \(|x|\). It tells us how far a number is from zero on a number line.

For instance, both -4 and 4 have an absolute value of 4, because each is four units away from zero. This is crucial for determining distance because distance, by its nature, cannot be negative. In our exercise, when we compute the distance between two points such as A and B, we subtract their coordinates and take the absolute value of the result.

• For \(|-6 - (-2)|\), the absolute value operation transforms \(-4\) into 4, which is the distance between points A and B.

Distance Formula

The Distance Formula on a number line is a specific application of the Absolute Value concept. It provides a clear method to calculate the space or distance between two points along a line.

The formula is quite simple: \[d(A, B) = |x_2 - x_1|\]Where \(x_1\) and \(x_2\) are the coordinates of two points. This formula emphasizes that distance should always be a positive value or zero, reflecting how far two points are without concern for direction.

For our exercise:

• The distance between A and C is calculated as \(|-6 - 4| = 10\).
• Following the formula ensures consistency in finding actual distances, regardless of which direction along the line we measure.

Number Line

A Number Line is a straight line on which every point represents a real number. It serves as a visual representation to help understand numbers, allowing us to order them, and visualize operations such as addition and subtraction as movements along the line.

In this context, the number line is a simple tool for visualizing the distances between points. Imagine points A, B, and C placed on this line at -6, -2, and 4, respectively.

• This helps us clearly see the placement and differences between the points.
• Using a number line, we can see that moving from -6 to -2 indeed covers a distance of 4 units, reaffirming our earlier calculations.

Understanding these concepts through the number line makes it easier to grasp distance relationships thematically, supporting more complex scenarios in higher dimensions as well.