Problem 75 - College Algebra Essentials

Question

In your own words, describe how to find the distance between two points in the rectangular coordinate system.

Step-by-Step Solution

Answer

The distance between two points in the rectangular coordinate system is calculated with the distance formula derived from the Pythagorean theorem: \[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \] where d is the distance, (x1, y1) are the coordinates of the first point and (x2, y2) are the coordinates of the second point.

Step 1: Understanding the Coordinate System

In a rectangular coordinate system, points are identified by coordinates (x, y). The x-coordinate represents the horizontal position and the y-coordinate represents the vertical position.

Step 2: Introduce the Distance Formula

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \] This is derived from the Pythagorean theorem as it calculates the hypotenuse of a right-angled triangle formed by the vertical and horizontal distances.

Step 3: Applying the Distance Formula

To find the distance between two points, subtract the x-coordinates of the two points and square the result, do the same for the y-coordinates, then add the two results together. The square root of this sum gives the distance between the two points.