Show that for points 𝐀(x1, 0) and 𝐁(x2, 0), d(𝐀, 𝐁)=|x2-x1|....

Name: Problem 27, Chapter 1: Modern Analytic Geometry
Uploaded: 2021-07-06T15:10:42-08:00
Duration: 1 min 11 s
Channel: Carson Merrill
Description: Show that for points 𝐀(x1, 0) and 𝐁(x2, 0), d(𝐀, 𝐁)=|x2-x1|.

Key Concepts

Euclidean Distance Formula

This is a method for determining the straight-line distance between two points in Euclidean space. The formula, derived from the Pythagorean Theorem, involves subtracting the coordinates of the points, squaring the differences, summing them, and then taking the square root. It provides a precise measure of distance in a coordinate system.

Pythagorean Theorem

This fundamental theorem of geometry relates the three sides of a right triangle. In the context of distance calculation, it justifies the use of the square root of the sum of the squares of the differences in coordinates. It is the underlying principle that leads to the formula used to calculate the distance between points.

Absolute Value Function

The absolute value function returns the non-negative magnitude of a real number. It is particularly useful in expressing distances since distance is always non-negative. When simplifying the Euclidean distance formula in one dimension, the square root of a square naturally results in the absolute value, ensuring a valid measure of distance.

Coordinate Geometry

This branch of geometry uses a coordinate system to describe and analyze the positions and relationships of points, lines, and figures. It enables the application of algebraic techniques to geometric problems, such as computing distances between points by representing them as coordinate pairs.

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