Problem 27 - Precalculus with Limits

Question

In Exercises 27-38, find the distance between the points. \( (6, -3) \), \( (6, 5) \)

Step-by-Step Solution

Answer

The distance between the points (6, -3) and (6, 5) is 8.

Step 1: Identifying Given Points

The two points given are (6, -3) and (6, 5). Identify these as (x1, y1) = (6, -3) and (x2, y2) = (6, 5) respectively.

Step 2: Simplifying the Distance Formula

Because the two points share the same x-coordinate, the distance formula simplifies to the absolute difference of the y-coordinates, i.e., |y2 - y1|. Substituting the given points values into the simplified formula gives: |-3 - 5|.

Step 3: Calculation

Calculate the absolute value of the expression |-3 - 5| to get: | -8 | which equals 8.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to determine the position of points in a plane using numerical coordinates. It's like giving every spot on a map a unique set of coordinates, called "ordered pairs," which look like (x, y). Each pair tells us the point's location on the x (horizontal) and y (vertical) axes. For example, the points given in the exercise are (6, -3) and (6, 5). Here, '6' is the x-coordinate, and it is the same for both points, which means they are vertically aligned on the coordinate plane.

• The x-coordinate tells us how far to move left or right.
• The y-coordinate tells us how far to move up or down.

In coordinate geometry, distances and angles can be measured between points, providing an understanding of the space each point occupies. This knowledge is fundamental in calculating distances using formulas.

Absolute Value

Absolute value is a mathematical function that tells us how far a number is from zero on a number line, regardless of direction. It is denoted by two vertical lines, like \(|x|\). The absolute value of a number is never negative.

• For example, both \(|8|\) and \(|-8|\) equal 8.
• Absolute value helps simplify calculations by removing negative signs.

In the context of the exercise, we use absolute value to find the distance between the y-coordinates of the two points. Since the x-coordinates are identical, subtract the y-values: \((-3) - (5) = -8\). The absolute value of \(-8\) is \(8\), which gives us the vertical distance between the two points.

Distance Calculation

Distance calculation in coordinate geometry often uses the distance formula, derived from the Pythagorean theorem. The general formula for calculating the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]However, when the points share the same x-coordinate, as in the given exercise, the formula simplifies as there's no horizontal distance:\[D = |y_2 - y_1|\]In our example, the points (6, -3) and (6, 5) only differ in their y-coordinates. Thus, we compute |(-3) - (5)|, resulting in \(8\). Here's why it's simplified:

• Simplification occurs because you're calculating a vertical line segment's length on the plane.
• The distance is a straightforward subtraction of y-coordinates, followed by calculating the absolute value.

This approach highlights how different types of coordinate alignments can affect the complexity of distance calculations in geometry.