Question

Length of a Line Segment Find the length of the line segment with the given endpoints. (-4,0) and (-6,0)

Short answer

The length of the line segment is 2 units.

Step-by-step solution

Understanding the Concept

The length of a line segment between two points can be calculated using the distance formula, which is derived from the Pythagorean theorem. However, because both points lie on a horizontal line (the y-coordinates are the same), the distance will be equal to the difference in the x-coordinates.

Subtract the X-Coordinates

To find the horizontal distance between the points (-4,0) and (-6,0), subtract the x-coordinate of the first point from the x-coordinate of the second point or vice versa, which will give us the absolute value of the distance. Distance = |x2 - x1| = |-6 - (-4)| = |-6 + 4| = |-2| = 2

Final Answer

The length of the line segment is 2 units.

Key concepts

Distance Formula

When it comes to understanding the distance between two points on a coordinate plane, the distance formula is a critical tool. It encapsulates the idea that any two points can have their separation measured in a standard, mathematical way. Based on the same principles as the Pythagorean theorem, this formula is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.

In more straightforward cases, like horizontal or vertical line segments, the distance formula simplifies significantly. If the line segment is horizontal (parallel to the x-axis), the y-coordinates will be the same, and the formula reduces to the absolute difference between the x-coordinates. Similarly, if the segment is vertical (parallel to the y-axis), the x-coordinates will be equal, and the distance is the absolute difference between the y-coordinates.

Pythagorean Theorem

At the heart of coordinate geometry lies the Pythagorean theorem, a cornerstone in mathematics, particularly when dealing with right triangles. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The formula is \( a^2 + b^2 = c^2 \).

This ancient theorem not only holds significance for geometric calculations but also lays the groundwork for the distance formula used in coordinate systems. Whenever you use the distance formula, you're applying the Pythagorean theorem to the 'triangle' formed by the change in the x and y coordinates between two points.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a system that utilizes algebraic equations to represent geometric figures. It's where algebra meets geometry, allowing complex shapes and lines to be expressed in a numerical format. In coordinate geometry, points are located on the plane using an ordered pair of numbers, known as coordinates, which define their position along the x (horizontal) and y (vertical) axes.

Understanding the concepts of the coordinate plane is essential when working with the distance formula or applying geometric principles like the Pythagorean theorem in a two-dimensional space. These applications enable us to solve problems involving distances, midpoints, gradients, and more, purely through algebraic methods.

Absolute Value

The absolute value of a number is a measure of its distance from zero on the number line, without considering which direction from zero the number lies. It is denoted by vertical bars, so the absolute value of \( x \) is written as \( |x| \). Essentially, the absolute value of a number is always non-negative.

This concept is particularly relevant when subtracting coordinates to find the length of a line segment. The actual distance can never be negative, as it represents the magnitude of separation between two points. Therefore, regardless of the order in which the points' coordinates are subtracted, taking the absolute value ensures the resulting distance is a positive number, accurately reflecting the true length of the line segment.