Question

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

Short answer

The function `distance` calculates the Euclidean distance between points \( (x_1, y_1) \) and \( (x_2, y_2) \) using the formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) and returns the result as type double.

Step-by-step solution

Understanding the distance formula

Recognize that the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the Cartesian coordinate system can be computed using the distance formula derived from the Pythagorean theorem. The formula is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Function setup

Begin by defining a function called `distance` that accepts four parameters representing the coordinates of the two points: `distance(double x1, double y1, double x2, double y2)`. This function will return a `double` which is the result of the distance computation.

Calculating the differences

Inside the function, calculate the differences between the x-coordinates \( (x_2 - x_1) \) and the y-coordinates \( (y_2 - y_1) \). Store these values in two separate variables, for example: `double dx = x2 - x1;` and `double dy = y2 - y1;`.

Applying the distance formula

Compute the square of the differences, add them, and then take the square root to find the distance. Use the C++ `std::pow` function for squaring and `std::sqrt` function for the square root. The calculation can be written as `double dist = std::sqrt(std::pow(dx, 2) + std::pow(dy, 2));`.

Returning the result

Finally, return the calculated distance `dist` from the function.

Key concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This relationship is expressed mathematically as \( c^2 = a^2 + b^2 \), where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides.

When it comes to calculating the distance between two points in a two-dimensional space, the Pythagorean theorem comes into play because the two points and the distance between them form a right-angled triangle. The horizontal and vertical distances between the points act as the perpendicular sides of the triangle, while the direct distance between them is the hypotenuse. Therefore, by using the Pythagorean theorem, one can calculate the hypotenuse — which in this case is the distance between the two points.

Cartesian Coordinate System

The Cartesian coordinate system is a two-dimensional plane defined by two perpendicular axes: the horizontal x-axis and the vertical y-axis. These axes intersect at a point called the origin, which has coordinates \( (0, 0) \). Any point in this plane can be specified by an ordered pair of numbers, \( (x, y) \) representing its distance from the origin along the x-axis and y-axis, respectively.

Using the Cartesian coordinate system makes it easy to visualize and calculate the positions and distances of points. When we calculate the distance between two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), we are essentially finding the length of the straight line that connects them. This process can be imagined as drawing a right triangle between the points, with the line segment as its hypotenuse, again invoking the Pythagorean theorem to find this length.

C++ Functions

In C++, a function is a self-contained block of code that performs a specific task. Functions allow for code reusability, modularity, and better organization of a program. In C++, a function is defined with a return type, a name, and a parameter list enclosed in parentheses. The function may return a value (using a return statement), or it may be void, indicating that it does not return a value.

The function \(distance\) described in the exercise is a perfect example of reusability and modularity. By encapsulating the distance calculation logic within a function, the code can be written once and then called multiple times with different pairs of points, simplifying the process of obtaining distances in a program. This function takes four \(double\) parameters representing the coordinates of the two points and returns a \(double\) value, which is the calculated distance. A key part of writing good functions in C++, or any programming language, is to ensure they perform their task well and are understandable to those who use or maintain the code.