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Chapter 2: Problem 11

The distance between two points \((x 1, y 1)\) and \((x 2, y 2)\) on a Cartesian coordinate plane is given by the equation $$ d=\sqrt{\left(x l-x^{2}\right)^{2}+(y l-y 2)^{2}} $$ Write a program to calculate the distance between any two points \((x 1, y 1)\) and \((x 2, y 2)\) specified by the user. Use good programming practices in your program. Use the program to calculate the distance between the points \((2,3)\) and \((8,-5)\).

Short Answer

Expert verified
The distance between the points \((2,3)\) and \((8,-5)\) is \(10\). A Python program to calculate this can be written as: ```python import math def distance(x1, y1, x2, y2): return math.sqrt((x1 - x2)**2 + (y1 - y2)**2) x1, y1 = 2, 3 x2, y2 = 8, -5 print("The distance between the points is:", distance(x1, y1, x2, y2)) ```

Step by step solution

01

Identify the given points

We are given two points as input: - Point 1: \((x_1, y_1) = (2, 3)\) - Point 2: \((x_2, y_2) = (8, -5)\)
02

Use the distance formula

Plug the given coordinates into the distance formula: $$ d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} = \sqrt{(2 - 8)^2 + (3 - (-5))^2} $$
03

Calculate the squared differences

Calculate the difference for the x-coordinate and y-coordinate, and square them: $$ (x_1 - x_2)^2 = (2 - 8)^2 = (-6)^2 = 36 $$ $$ (y_1 - y_2)^2 = (3 - (-5))^2 = (3 + 5)^2 = 8^2 = 64 $$
04

Add the squared differences

Add the squared differences together: $$ 36 + 64 = 100 $$
05

Calculate the square root

Determine the square root of the sum: $$ d = \sqrt{100} = 10 $$
06

Write a program to calculate the distance

A simple Python program to calculate the distance between the given points could look like this: ```python import math def distance(x1, y1, x2, y2): return math.sqrt((x1 - x2)**2 + (y1 - y2)**2) x1, y1 = 2, 3 x2, y2 = 8, -5 print("The distance between the points is:", distance(x1, y1, x2, y2)) ``` In this program, we define a function called `distance` that takes the coordinates of two points as input and returns the distance between them using the provided formula. We then use the example points \((2,3)\) and \((8,-5)\) to calculate the distance and print the result. #Conclusion# The distance between the points \((2,3)\) and \((8,-5)\) is \(10\).

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