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Chapter 11: Problem 127

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

Short Answer

Expert verified
The distance between two points (A,B) in a rectangular coordinate system can be found by using the distance formula: \(d = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}}\). Each point is represented by coordinates (x, y). Substitute the known coordinates into the formula, calculate the difference between the x's and y's, square those differences, add the squares, and take the square root of the sum to find the distance between the points.

Step by step solution

01

Understand the Concept of Coordinate System

Firstly, it is important to understand the concept of a rectangular or Cartesian coordinate system. It is a form of graph that allows every location in a two-dimensional space to be specified by a set of numerical coordinates, which are the distances to the graph from two fixed perpendicular directed lines, expressed in the same unit of length.
02

Identify the Coordinates

Then, identify the coordinates of the two points. Each point in the rectangular coordinate system is represented by an ordered pair of numbers (x, y). For instance, for two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\), the coordinates \(x_1, y_1\) are for point A and the coordinates \(x_2, y_2\) are for point B.
03

Apply the Distance Formula

After identifying the coordinates, apply the distance formula. The distance \(d\) between two points in the rectangular coordinate system is given by the formula \(d = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}}\). This formula is derived from Pythagoras' theorem, and it calculates the 'direct' distance 'as the crow flies', between the two points.
04

Substitute the values and solve

Now, substitute the given coordinates into the distance formula. Perform the operation inside the parentheses first (x_2 - x_1 and y_2 - y_1), square the results, add the two resulting numbers, and then take the square root of the resultant sum to get the distance.

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