Problem 16 - 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) \(d(A, C)\) $$ -6,-2,4 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Points and Distances

First, identify the coordinates assigned to points \(A\), \(B\), and \(C\). Given coordinates are -6, -2, and 4, respectively. Therefore, point \(A\) has a coordinate of -6, \(B\) has -2, and \(C\) has 4. The exercise asks for the distances between these points: \( A\) and \( B\), \( B\) and \( C\), \(C\) and \(B\), and \(A\) and \(C\).

Step 2: Calculate Distance d(A, B)

To find the distance \(d(A, B)\), use the formula \( |b - a| \) for the distance between two points on a line, where \(a\) and \(b\) are the coordinates of \(A\) and \(B\). Here, \(a = -6\) and \(b = -2\). Thus, \(d(A, B) = |-2 - (-6)| = |-2 + 6| = |4| = 4\).

Step 3: Calculate Distance d(B, C)

Next, calculate the distance \(d(B, C)\) using the same distance formula. Here \(b = -2\) and \(c = 4\). So, \(d(B, C) = |4 - (-2)| = |4 + 2| = |6| = 6\).

Step 4: Calculate Distance d(C, B)

The distance \(d(C, B)\) is the same as \(d(B, C)\) since distance is a non-negative quantity and symmetric, meaning \(d(B, C) = d(C, B)\). Therefore, \(d(C, B) = 6\).

Step 5: Calculate Distance d(A, C)

Finally, calculate \(d(A, C)\) using the coordinates \( a = -6 \) and \(c = 4\). So, \(d(A, C) = |4 - (-6)| = |4 + 6| = |10| = 10\).

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of geometry where we use a coordinate system to analyze various geometric shapes and figures. For number line problems, each point corresponds to a coordinate on the line. The number line is a straight line where each point on it represents a real number.
In this exercise, the points A, B, and C are given as coordinates on a number line, specifically at -6, -2, and 4 respectively.

• Point A represents the value -6.
• Point B represents the value -2.
• Point C represents the value 4.

Using these coordinates, we apply simple arithmetic to determine distances between the points. This is a basic application of coordinate geometry, helping visualize and calculate spatial relationships on the number line.

Distance Formula

The distance formula helps us find out how far apart two points are on a coordinate line. This formula is essential when working with points on a number line, making it easy to calculate the distance between any two points.
The formula for distance between two points with coordinates, say \(x_1\) and \(x_2\), on a number line is: \[d = |x_2 - x_1|\] This formula makes use of absolute values to ensure that the distance is always a non-negative value. In our exercise:

• The distance between points A (-6) and B (-2) calculated as \(|-2 - (-6)|\) results in 4 units.
• To find the distance between B (-2) and C (4), we calculate \(|4 - (-2)|\) leading to 6 units.
• For A (-6) to C (4), the distance is \(|4 - (-6)|\), which equals 10 units.

By using the distance formula, calculations become systematic and straightforward.

Absolute Value

Absolute value is a concept that simplifies the idea of distance as a non-negative number. It represents the magnitude of a value, regardless of its direction on a number line.
For example, the absolute value of -4 is 4, represented by the notation \(|-4| = 4\). This is because the distance is always positive, whether measured forward or backward from a point.
In this exercise, the distances:

• From A to B (\(|-2 + 6| = |4| = 4\))
• From B to C (\(|4 + 2| = |6| = 6\))
• From A to C (\(|4 + 6| = |10| = 10\))

showcase how we utilize absolute values to ensure that the answer remains positive, reinforcing the intuitive meaning of distance as an always non-negative measure.

Symmetry of Distance

Symmetry in distance refers to the fact that the distance between two points remains the same no matter which point is the starting point or the endpoint. This trait is particularly handy when solving problems involving distances.
Using our exercise as an example:

• The distance from B to C, \(d(B, C) = 6\), is identical to the distance from C to B, \(d(C, B) = 6\). This demonstrates the principle of symmetry in measuring distances.
• This holds for any two points on a number line indicating a universal property of distances in geometry.

Symmetry simplifies calculations and teaches that direction does not matter when computing how far apart two points are. It's a symmetry of distance that often comes into play in coordinate geometry tasks.