Problem 18 - Technical Mathematics with Calculus

Question

Length of a Line Segment Find the length of the line segment with the given endpoints. (-2.06,-5.83) and (-2.06,-8.34)

Step-by-Step Solution

Answer

The length of the line segment is 2.51 units.

Step 1: Identify the endpoints

The endpoints of the line segment are given as A(-2.06, -5.83) and B(-2.06, -8.34). Note that the x-coordinates of the two points are the same.

Step 2: Apply the distance formula

Since the x-coordinates are the same, we can find the length of the line segment by calculating the difference between the y-coordinates. The length, or distance d, between two points with the same x-coordinate is given by the formula: \( d = |y_2 - y_1| \).

Step 3: Calculate the length of the segment

Substitute the y-coordinates of both points into the formula: \( d = |-8.34 - (-5.83)| = |-8.34 + 5.83| = |-2.51| = 2.51 \). Therefore, the length of the line segment is 2.51 units.

Key Concepts

Distance Formula

The distance formula is crucial for determining the space between two points on a Cartesian plane. It is an application of the Pythagorean theorem from geometry.

For two points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) the formula is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). However, when the x-coordinates (or y-coordinates) of both points are the same, like in our exercise, the distance is the absolute difference of the y-coordinates (or x-coordinates).

In simpler terms, the formula shows that the direct path connecting two points is the straightest, and thereby the shortest, distance that can be measured between them. Remembering that the distance is always a positive quantity is crucial, as a line segment's length can't be negative.

Cartesian Coordinate System

The Cartesian coordinate system is a two-dimensional planar system used for determining the precise location of points. The system consists of two number lines, the x-axis (horizontal) and y-axis (vertical), that intersect at a right angle at the origin, marked as \( O(0, 0) \).

Every point in this system is defined by an ordered pair of numbers \( (x, y) \) that correspond to the horizontal (x) and the vertical (y) distances from the origin. By using this system, you can visualize geometric shapes, calculate areas, lengths, and other geometrical properties. It's fundamental to much of geometry and algebra.

Absolute Value

The absolute value of a number is simply its distance from zero on the number line, irrespective of direction. Denoted by vertical bars \( |x| \) for any number \( x \), this concept is key in understanding that distance is always non-negative.

For example, the absolute value of both \( 3 \) and \( -3 \) is \( 3 \) because both are three units away from zero on the number line. In the context of our exercise, the absolute value converts any negative difference in coordinates into a positive length, ensuring that the segment's length is correctly represented as a positive value.

Geometry in Mathematics

Geometry is a branch of mathematics concerned with the properties and relationships of points, lines, angles, surfaces, and shapes. It serves as a foundational component in many areas of mathematics and practical applications such as engineering, architecture, and physics.

In our exercise, geometry's role is evident in determining the length of a line segment, which is one of the simplest forms of geometric figures. Understanding fundamental geometric principles is crucial for tackling more complex problems and develops spatial reasoning and problem-solving abilities.