Question

Define a function hypotenuse that calculates the hypotenuse of a right triangle when the other two sides are given. The function should take two double arguments Define a function hypotenuse that calculates the hypotenuse of a right triangle when the other two sides are given. The function should take two double arguments $$\begin{array}{lll} \text { Triangle } & \text { Side I } & \text { Side 2 } \\ \hline 1 & 3.0 & 4.0 \\ 2 & 5.0 & 12.0 \\ 3 & 8.0 & 15.0 \end{array}$$

Short answer

Using the Pythagorean Theorem, the hypotenuses for the triangles are, respectively, 5.0 for Triangle 1, 13.0 for Triangle 2, and 17.0 for Triangle 3.

Step-by-step solution

Understanding the Pythagorean Theorem

The Pythagorean Theorem states that for a right triangle with sides of length 'a' and 'b', and hypotenuse 'c', the equation is given by \(c^2 = a^2 + b^2\). This will be the basis for the 'hypotenuse' function.

Define the Function

The 'hypotenuse' function needs to take two double arguments, 'side1' and 'side2'. It should calculate the hypotenuse length 'c' by taking the square root of the sum of the squares of 'side1' and 'side2' (\(c = \sqrt{side1^2 + side2^2}\)).

Calculating Hypotenuse for Each Triangle

For each triangle given, input the lengths of Side 1 and Side 2 into the 'hypotenuse' function. For example, for Triangle 1 with sides 3.0 and 4.0, the hypotenuse is calculated as \(\sqrt{3.0^2 + 4.0^2}\), which simplifies to \(\sqrt{9.0 + 16.0}\) or \(\sqrt{25.0}\), which is 5.0.

Key concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, particularly relevant in the study of right triangles. 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 can be expressed mathematically as:
c^2 = a^2 + b^2,
where c is the length of the hypotenuse, while a and b are the lengths of the triangle's other two sides. Understanding this theorem is crucial since it forms the underpinning of numerous mathematical, scientific, and engineering calculations, including the development of the hypotenuse function in C++ programming.

Function Definition

In C++ programming, a function is a block of code designed to perform a specific task. A function definition includes the function's name, its return type, and parameters (if any). It also contains the body of the function—where the code statements that define the function's behavior are located.

Basic Structure

When creating a function to calculate the hypotenuse, you might start with a structure like this:

double hypotenuse(double side1, double side2) {   // Function body}

Here, double signifies the return type, hypotenuse is the name, and side1 and side2 are the parameters the function will use to calculate the hypotenuse. The body of the function is where the logic of the Pythagorean Theorem is applied to return the length of the hypotenuse.

Double Data Type

The double type in C++ is a data type used to represent floating-point numbers, which are numbers that have a decimal point. Doubles provide more precision than float types and can store larger numbers, making them well-suited for mathematical calculations where accuracy is crucial.
When defining a hypotenuse function, we use double to ensure that the calculation can handle decimals and provide a precise result. For example, the sides of a right triangle might be measured in meters with several decimal points, requiring that level of detail in computations.

Square Root Calculation

Calculating the square root of a number is a common operation in mathematics, and it's directly used in the implementation of the Pythagorean Theorem when solving for the hypotenuse. In C++, this operation is carried out using the sqrt() function, which is part of the header file. When finding the hypotenuse, after squaring the two sides and adding them together, you would use sqrt() to determine the length of the hypotenuse:

double hypotenuse(double side1, double side2) {    return sqrt(side1 * side1 + side2 * side2);}

The function sqrt() calculates the square root of the expression side1 * side1 + side2 * side2, thus completing the computation as dictated by the Pythagorean Theorem.

C++ Functions and Arguments

In C++, a function can accept inputs known as arguments that influence the function's execution. For the hypotenuse function, the arguments are the lengths of the two sides of a right triangle (as double values). They are passed to the function when it's called, allowing it to execute with those specific values.

• Defining Arguments: Within the function definition, the arguments are the variables listed in parentheses after the function name.
• Passing Arguments: When calling the function, you provide the actual values for these variables.

By using arguments effectively, a single function can perform its task using different sets of data, which in this case, allows the hypotenuse function to be reused to calculate the hypotenuse for any set of triangle sides provided.