Question

Write a program that reads three nonzero integers and determines and prints whether they could be the sides of a right triangle.

Short answer

Input three integers, assign the largest as hypotenuse, check using the Pythagorean theorem.

Step-by-step solution

Input the integers

The first step is to gather input from the user. We need to read three nonzero integers, which will be the potential sides of a triangle. We can store them in variables, for example, `a`, `b`, and `c`.

Check nonzero condition

Ensure that all integers are nonzero. This step involves checking if any of the integers `a`, `b`, or `c` are zero. If any is zero, the program should notify the user and stop further execution as zero cannot represent a triangle side.

Identify potential hypotenuse

For any right triangle, the hypotenuse is the largest side. We need to determine which of the integers is the largest and assign it to the variable `c`, and the other two as `a` and `b`. This can be done using conditional checks to swap values if needed so that `c` will be the largest.

Check right triangle condition

Now, check if the integers satisfy the Pythagorean theorem: \[ a^2 + b^2 = c^2 \]This condition ensures that the triangle is a right triangle if true. Compute both sides of the equation and compare them for equality.

Output the result

Finally, based on whether the Pythagorean theorem holds, print the appropriate message. If it is satisfied, print 'The integers form a right triangle.' Otherwise, print 'The integers do not form a right triangle.'.

Key concepts

Control Structures in C++

In C++ programming, control structures are essential for directing the flow of the program. They allow code to take decisions, loop through a set of elements, or jump between different code parts. Control structures mainly include `if` statements, loops like `for` and `while`, and `switch` cases.

For the given exercise, control structures help in evaluating conditions and branching the code flow according to those conditions. This involves selecting the correct path in the program — like determining whether the given integers qualify to form a right triangle. By using control structures, you can control how the program processes information and executes commands.

Conditional Statements

Conditional statements in C++ are fundamental when making decisions in the code. They include statements like `if`, `else if`, and `else`. These allow the program to choose different actions based on conditions.

For the task of determining if three numbers can make a right triangle, conditional statements check various criteria:

• Whether any of the integers is zero — using an `if` statement to alert the user if they are zero.
• Selecting the largest of the three numbers to be considered as the hypotenuse — using `if` conditions to evaluate which number is largest.
• Verifying the condition of the Pythagorean theorem — putting these checks within the `if` statements to decide if \( a^2 + b^2 = c^2 \) holds true.

Through these conditions, you can direct the program to appropriately let the user know if the numbers form a right triangle or not.

User Input Handling in C++

Handling user input in C++ is crucial in creating interactive programs. User input is typically gathered using functions like `cin`. This allows the user to input data which the program can use to perform actions or calculations.

In the exercise, capturing three integers from the user is the starting point. It involves reading input values and storing them in variables, say `a`, `b`, and `c`. Properly handling input involves validating that these numbers are nonzero and alerting the user for incorrect inputs. This process ensures that the program only processes valid information, necessary for determining if these numbers can form the sides of a triangle.

• `cin` is used for reading data input.
• We validate the input to ensure all numbers are distinct and non-zero, safeguarding the program from invalid states.

Mastering user input handling makes programs robust and user-friendly.

Pythagorean Theorem Application

The Pythagorean Theorem is a geometric principle used to relate the sides of a right triangle. It states that the square of the hypotenuse (largest side) is equal to the sum of squares of the other two sides. Mathematically, it is expressed as: \[ a^2 + b^2 = c^2 \] where `c` is the hypotenuse and `a` and `b` are the other sides.

In this exercise, applying the Pythagorean theorem requires computing squares of the integers and checking if the equation holds. This involves:

• First, identifying the largest integer to use as the hypotenuse.
• Calculating the squares of each side.
• Verifying if the sum of squares of shorter sides equals the square of the hypotenuse.

By validating this condition, the program can determine if the integers can form a right triangle and inform the user. This demonstrates a practical use of mathematics within programming and problem-solving.