Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 15-18: The given numbers are coordinates of points \(\boldsymbol{A}, \boldsymbol{B}\), and \(\boldsymbol{C}\), respectively, on a coordinate line. Find the distance. (a) \(d(A, B)\) (b) \(d(\boldsymbol{B}, \boldsymbol{C})\) (c) \(d(\boldsymbol{C}, \boldsymbol{B})\) (d) \(d(A, C)\) $$ -9,1,10 $$

Step-by-Step Solution

Answer

(a) 10, (b) 9, (c) 9, (d) 19

Step 1: Understand the Point Coordinates

The coordinates for point A, B, and C are given as real numbers on a number line. Specifically: \( \boldsymbol{A} = -9 \), \( \boldsymbol{B} = 1 \), and \( \boldsymbol{C} = 10 \). These numbers represent positions on a coordinate line.

Step 2: Formula for Distance on a Coordinate Line

The distance between two points \(x\) and \(y\) on a coordinate line is calculated using the formula \( d(x, y) = |x - y| \). This formula represents the absolute value of their difference.

Step 3: Calculate the Distance from A to B

Using the formula, the distance from \(A\) to \(B\) is: \[ d(A, B) = |-9 - 1| = |-10| = 10 \]

Step 4: Calculate the Distance from B to C

The distance from \(B\) to \(C\) using the distance formula is: \[ d(B, C) = |1 - 10| = |-9| = 9 \]

Step 5: Calculate the Distance from C to B

Since distance is a measure and not directional, \( d(C, B) = d(B, C) \). Therefore: \[ d(C, B) = |10 - 1| = |9| = 9 \]

Step 6: Calculate the Distance from A to C

Finally, the distance from \(A\) to \(C\) is: \[ d(A, C) = |-9 - 10| = |-19| = 19 \]

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses a coordinate system to study geometrical shapes and figures. In this system, every point is defined by a set of numerical coordinates which are determined by the intersection of the axes.

In a one-dimensional space such as a number line, every point is represented by a single coordinate. In this exercise, the coordinates -9, 1, and 10 denote the positions of points A, B, and C, respectively. These numbers serve to describe specific locations along the number line.

The beauty of coordinate geometry lies in its ability to translate geometrical problems into algebraic equations, making it a powerful tool for solving a wide range of mathematical problems.

Number Line

A number line is a straight line on which every point corresponds to a real number, and these numbers are spaced equally along the line. It provides a visual representation of numbers in order, from left to right.

On this line:

• Negative numbers are located to the left of zero.
• Positive numbers are located to the right of zero.
• Distance between any two numbers is consistent throughout.

For the given exercise, the points -9, 1, and 10 are positioned on this number line. This representation helps in understanding and calculating the distances between these points efficiently.

The number line also helps in visualizing operations like subtraction as moving to the left along the line, which is particularly useful in understanding the concept of distance.

Absolute Value

The absolute value of a real number is the distance between that number and zero on a number line, regardless of direction. It is always a non-negative number.

Mathematically, the absolute value of a number \( x \) is denoted as \( |x| \), and it is defined as:

• \( |x| = x \) if \( x \geq 0 \)
• \( |x| = -x \) if \( x < 0 \)

In the exercise, the distance between any two points \( x \) and \( y \) is calculated using the formula \( d(x, y) = |x - y| \).

This formula highlights how absolute value converts the difference between two numbers into its non-negative equivalent, symbolizing the concept that distance is always a positive measurement, representing the length between two points.

Real Numbers

Real numbers include all the numbers on the number line, covering both rational and irrational numbers. They encompass all the possible values a coordinate can take along the number line.

Here are some important aspects of real numbers:

• They can be integers, like -9 and 1, which make up part of our points A and B in this exercise.
• They can also be decimal numbers or fractions, filling all the "gaps" between integers.

Real numbers thus form the continuous line of values that points A, B, and C can take. This concept is fundamental in coordinate geometry because any real number can be a coordinate point on a number line.

Understanding real numbers helps to grasp the idea that any point can fall anywhere along the number line, a crucial concept for comprehending distances and their calculations.