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Chapter 1: Problem 18

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. \(d(A, C)\) $$8 .-4 .-1$$

Short Answer

Expert verified
A: 12, B: 3, C: 3, D: 9

Step by step solution

01

Identify Coordinates

First, identify the given coordinates for points A, B, and C. We have: \( A = 8 \), \( B = -4 \), and \( C = -1 \).
02

Calculate Distance d(A, B)

To find the distance between points A and B, use the formula \( d(A, B) = |A - B| = |8 - (-4)| = |8 + 4| = 12 \).
03

Calculate Distance d(B, C)

To find the distance between points B and C, use the formula \( d(B, C) = |B - C| = |-4 - (-1)| = |-4 + 1| = 3 \).
04

Calculate Distance d(C, B)

Notice that the distance \( d(C, B) \) is the same as \( d(B, C) \) because distance is always positive, so \( d(C, B) = 3 \).
05

Calculate Distance d(A, C)

To find the distance between points A and C, use the formula \( d(A, C) = |A - C| = |8 - (-1)| = |8 + 1| = 9 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a branch of mathematics that combines algebra and geometry. It allows us to represent geometric figures on a coordinate plane using numerical coordinates. This makes it simpler to compute properties like distance, midpoint, and angles between different geometric objects.
  • In this exercise, points are represented as coordinates on a number line. For example, the point A on a number line is represented by the number 8.
  • The number line can be thought of as a simplified version of a coordinate plane, where each point has a single coordinate value.
By expressing these points as coordinates, we can easily calculate distances and other properties using straightforward algebraic formulas, which are foundational in coordinate geometry.
Absolute Value
The absolute value of a number refers to its distance from zero on a number line. Essentially, it's the number without regard to its sign, whether positive or negative.
  • In mathematical terms, the absolute value of a number \( x \) is denoted as \( |x| \).
  • This is crucial when calculating distance, as distance should always be a non-negative value.
For example, the absolute value of \(-4\) is \(|-4| = 4\). When calculating distances between points A, B, and C, the absolute value ensures we correctly compute the length of the interval between any two points, without concerning ourselves with direction.
Distance Formula
The distance formula is a mathematical equation used to determine the length between two points on a line. When working with a coordinate line or a number line, this formula simplifies to calculating the absolute difference between the two given coordinates.
  • The formula to calculate the distance \( d \) between two points \( X \) and \( Y \) is \( d(X, Y) = |X - Y| \).
  • This formula is derived from the concept of absolute value. It simply takes the difference of two coordinates and determines how far apart they are.
In our exercise, you can see this formula applied repeatedly to find distances like \( d(A, B) = |8 - (-4)| \), which results in 12 units. The formula remains effective because it accounts only for the magnitude of separation, leading to a consistent measure of distance.
Points on a Line
When we talk about points on a line, particularly a coordinate line or number line, each point corresponds to a single numerical value representing its position. This concept is integral for simplifying complex geometrical problems to more basic arithmetic calculations.
  • Each point is a unique value that can be positive, negative, or zero, as seen in our example with points A, B, and C having coordinates 8, -4, and -1.
  • Finding the distance between these points involves understanding that each point represents a specific location on the number line.
By visualizing these points and understanding their positions, one can easily calculate the distances needed by using their coordinates. This straightforward visualization technique makes it easy to understand and solve distance problems efficiently.

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Most popular questions from this chapter

Simplify the expression, assuming \(x\) and \(y\) may be negative. b$$\sqrt[4]{(x+2)^{12} y^{4}}$$

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