Question

Write function distance that calculates the distance between two points \((x 1, y 1)\) and \((x 2,\) \(y 2\) ). All numbers and return values should be of type double.

Short answer

Use the distance formula \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\) to implement the function.

Step-by-step solution

Understand the Problem

To calculate the distance between two points \(x_1, y_1\) and \(x_2, y_2\), we can use the distance formula. The problem requires us to write a function that accepts these coordinates as input, computes the distance, and returns it.

Analyze the Distance Formula

The distance formula to find the distance between two points \(x_1, y_1\) and \(x_2, y_2\) on a 2D plane is given by: \[ \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \]. This formula is derived from the Pythagorean theorem.

Write the Function Signature

The function should be designed to take four double parameters—representing the coordinates \(x_1, y_1, x_2,\) and \(y_2\)—and return a double value.

Implement the Function

Using a programming language like Python, you can implement the function "distance" like this: ```python def distance(x1, y1, x2, y2): return ((x2 - x1) ** 2 + (y2 - y1) ** 2) ** 0.5 ``` This code correctly calculates and returns the distance based on the formula derived earlier.

Key concepts

Distance Formula

The distance formula is a mathematical equation used to determine the distance between two points in a Cartesian coordinate system. It is particularly useful in a 2D plane. The formula is expressed as follows:\[ \text{distance} = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \]This formula is derived from the concept of measuring the hypotenuse of a right triangle when the two points are considered as the opposite corners of the rectangle. It helps compute the shortest distance directly between the two points, often used in geometry, physics, and computer programming. The distance is typically calculated in the same unit as the coordinates. Therefore, it is crucial to ensure that the coordinates are in the same units before using this formula.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that establishes a relationship between the sides of a right triangle. It states that the sum of the squares of the two shorter sides is equal to the square of the hypotenuse, which is the side opposite the right angle. The theorem can be written as:\[ a^2 + b^2 = c^2 \]where \(a\) and \(b\) are the lengths of the shorter two sides, and \(c\) is the length of the hypotenuse. This theorem is the basis for the distance formula. When you view the change in \(x\) coordinates as one leg and the change in \(y\) coordinates as another leg of a right triangle, the distance between two points is the hypotenuse. Thus, the distance formula is essentially an application of the Pythagorean theorem to find the length of that hypotenuse in a 2D coordinate system.

Function Implementation

Function implementation in programming involves defining a section of reusable code that performs a specific task. In our case, we want to implement a function that calculates distance using the distance formula. When implementing a function in C++ or a similar programming language, follow these typical steps:- Define the function's return type. For distance calculation, this would be a double since distances can be decimal numbers.- Specify the function name, for example, `distance`.- Declare the parameters required. For this task, we need four doubles corresponding to the coordinates \(x_1, y_1, x_2,\) and \(y_2\).- Inside the function, apply the distance formula.- Return the final result.Here is a C++ example:```cpp#include double distance(double x1, double y1, double x2, double y2) { return std::sqrt((x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1));}```In this function, we use `std::sqrt` from the cmath library to compute the square root, an essential part of the distance formula.

2D Coordinate Geometry

2D coordinate geometry is a branch of mathematics that deals with points, lines, and shapes on a two-dimensional plane. Each point in this plane is defined by a pair of numerical coordinates. These coordinates are expressed as \((x, y)\), representing the point's horizontal (x) and vertical (y) positions relative to the origin of the plane.In practical applications:- The x-axis runs horizontally across the plane, while the y-axis runs vertically.- Together, they form a grid on which points can be plotted. The intersection of these axes is the origin, represented by the coordinates (0, 0).- The distance between any two such points can be measured using the distance formula.Understanding 2D coordinate geometry is crucial for fields like computer graphics, engineering, and physics. It provides foundational knowledge needed for visualizing how objects and points are situated relative to each other.