Question

(Sides of a Right Triangle) Write a program that reads three nonzero integers and determines and prints whether they're the sides of a right triangle.

Short answer

To check if three sides of a triangle form a right triangle, identify the longest side as the hypotenuse and the remaining sides as legs, then verify if the sum of the squares of the legs equals the square of the hypotenuse.

Step-by-step solution

Understand the Problem

We are tasked to create a program that takes three nonzero integers as input. These integers represent the sides of a triangle. The goal is to determine whether these sides can form a right triangle. A right triangle is one where the square of the length of the longest side (the hypotenuse) is equal to the sum of the squares of the other two sides (Pythagorean theorem).

Read Input

First, the program will prompt the user to enter three nonzero integers. For example, suppose the user enters 3, 4, and 5. These will be assigned to variables, typically called a, b, c or similar identifiers.

Identify the Hypotenuse

To apply the Pythagorean theorem, identify the longest side, which is the hypotenuse. The two other sides are the legs. Amongst the entered sides, find the maximum value and treat it as the hypotenuse. The other two sides will be the legs.

Apply the Pythagorean Theorem

Check if the squares of the two legs add up to the square of the hypotenuse. In mathematical terms, this check is expressed as a^2 + b^2 = c^2, where c is the hypotenuse and a, b are the other two sides.

Print the Result

If the condition from the Pythagorean theorem is satisfied, print a message indicating that the integers can be the sides of a right triangle. If the condition is not satisfied, print a message that indicates they cannot form a right triangle.

Key concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry, specifically dealing with right triangles. It states that for any right 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 other two sides. This can be represented by the equation \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse and \( a \) and \( b \) are the other two sides.

Understanding this theorem is crucial, not only in geometry but also in various fields such as architecture, physics, and computer science. In programming, the Pythagorean theorem can be used to verify if a set of three given sides can form a right triangle, which is vital for the validation program in question.

In our context, once the user inputs three integers representing the sides of a triangle, the program uses the Pythagorean theorem to verify the possibility of these sides forming a right triangle. The program squares the lengths of the two shorter sides and checks if their sum is equal to the square of the longest side. This mathematical validation is an excellent example of how a centuries-old theorem still finds relevance in modern computing applications.

C++ control structures

In C++, control structures are used to dictate the flow of a program's execution, allowing the program to make decisions or repeat operations. These structures include conditionals like \( if \) and \( else \) statements, as well as loops like \( for \) and \( while \).

For our right triangle validation program, control structures lead the execution path. Once the input is received, the program employs an \( if \) statement to determine if the provided sides adhere to the Pythagorean theorem. Based on this conditional check, different outputs are presented to the user.

Using Control Structures:

Let's say variable \( c \) is the presumed hypotenuse. The program examines \( a^2 + b^2 \) versus \( c^2 \). If they are equal, the \( if \) block executes a print statement to confirm a right triangle. If they are not equal, an \( else \) block handles the output, denying the possibility of a right triangle. Thus, C++ control structures play a pivotal role in the logical progression and output presentation of the program.

Programming logic

Programming logic involves the application of logical reasoning to solve problems using a sequence of instructions. It's the foundation of writing efficient and effective code. For our program that determines if three sides can form a right triangle, the logic is rooted in understanding the Pythagorean theorem and implementing it correctly within the code.

Logical steps in the program begin with input validation, ensuring that the supplied numbers are non-zero integers. The next step is identifying the largest number as the potential hypotenuse. This is followed by checking if the relationship between the sides fulfills the Pythagorean theorem's criteria. Finally, based on this logical evaluation, the program provides feedback to the user.

Effective programming logic seeks to minimize complexity while enhancing readability and performance. In our case, the logic should be structured so that it leads to minimal computation, avoiding unnecessary checks or iterations, and delivering fast and accurate results to the end-user.

Conditional statements

Conditional statements in programming are used to perform different actions based on different conditions. In C++, these are primarily the \( if \) and \( else \) statements. They're pivotal in deciding how a program behaves under various inputs or other state-related situations.

When the right triangle validation program receives its input, a series of conditional statements take over. The core condition inspects whether \( a^2 + b^2 == c^2 \), determining the triangle's validity as a right triangle. The conditional block that corresponds to 'true' will output that the sides can indeed form a right triangle, while the 'false' block will state the opposite.

Efficiency in Conditionals:

The program should also make efficient use of conditionals to avoid excessive or redundant calculations. For instance, once the hypotenuse is identified, no further comparisons are necessary. This smart use of conditional statements directly impacts the program's performance and reliability while ensuring that the logic remains clear and manageable.