Problem 12 - Technical Mathematics with Calculus

Question

Length of a Line Segment Find the length of the line segment with the given endpoints. (0,3) and (0,-5)

Step-by-Step Solution

Answer

The length of the line segment is 8 units.

Step 1: Identify the endpoints

The endpoints of the line segment are given as A(0, 3) and B(0, -5).

Step 2: Use the distance formula

Since the points have the same x-coordinate, we can calculate the distance by subtracting the y-coordinates: Length of line segment = |y2 - y1|. Where (0,3) is (x1, y1) and (0,-5) is (x2, y2).

Step 3: Subtract the y-coordinates

Subtract y1 from y2 to find the length: Length = |-5 - 3|.

Step 4: Calculate the absolute value

The length of the line segment is the absolute value of the difference, so Length = |-(5 + 3)| = |-8| = 8.

Key Concepts

Distance Formula

Understanding the distance formula is essential in coordinate geometry when calculating the length of a line segment. The distance formula \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \) is derived from the Pythagorean theorem and measures the straight line distance between two points in the Cartesian coordinate system.

For vertical or horizontal line segments, this formula simplifies significantly because one set of coordinates are the same. In our exercise, the x-coordinates are identical, which means that the distance can be found by calculating the absolute difference of the y-coordinates, rendering the use of the square root unnecessary. Thus, the length of the line segment is \( |y_2 - y_1| \) which leads to a simpler calculation.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, provides a link between algebra and geometry through the use of coordinates. It utilizes numerical methods to represent and solve geometrical problems. The coordinates of points are used to find the length of line segments, the slopes of lines, and the equations of lines among other properties.

In the given example, the line segment has been expressed through coordinates (0,3) and (0,-5). These coordinates indicate that the line segment is vertical and therefore we only need to consider the y-coordinates to find the distance. Understanding the graphing of points and the x and y axes in a plane are fundamental concepts of coordinate geometry that lead to solving many other complex problems.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is represented by two vertical bars, such as \( |x| \). The absolute value turns negative numbers into positives, since distance cannot be negative. In geometrical terms, this concept ensures that the lengths of line segments are always expressed as positive numbers, because a segment's length can never be negative.

In the step by step solution of our exercise, the absolute value is used to compute the length of the line segment after finding the difference of the y-values. When calculating \( |-8| \), we are concerned with the distance from 0, so the result is a positive 8, reflecting the positive nature of length in geometry.