Question

Write an application that reads three nonzero integers and determines and prints whether they could represent the sides of a right triangle.

Short answer

To determine if three sides form a right triangle, identify the largest side as the potential hypotenuse, and check if the sum of the squares of the other two sides equals the square of the hypotenuse (Pythagorean theorem).

Step-by-step solution

- Read the Integers

Prompt the user to input three nonzero integers. Store these integers in variables, labeling them as side1, side2, and side3. It's important to verify that each input is a nonzero integer.

- Identify the Hypotenuse

Determine which of the three sides is the longest, as this side would be the hypotenuse if the sides form a right triangle. Assign this longest side to a variable called 'hypotenuse', and the other two sides to 'sideA' and 'sideB'.

- Apply the Pythagorean Theorem

Use the Pythagorean theorem, which states that in a 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 lengths of the other two sides. Check if hypotenuse^2 is equal to sideA^2 + sideB^2 (making sure to square the lengths).

- Determine if the Sides Form a Right Triangle

If the equality from Step 3 holds true, then print a message indicating the integers can form a right triangle. If not, print a message saying the integers cannot form a right triangle.

Key concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that provides a relation between the three sides of a right triangle. This theorem can be expressed with the equation: \( a^2 + b^2 = c^2 \) where \( a \) and \( b \) represent the lengths of the legs of the triangle, and \( c \) is the length of the hypotenuse, the longest side opposite the right angle. The theorem only applies to right triangles, and it's a reliable method to test if a triangle with given side lengths is a right angle triangle.

For the classroom problem, we use the Pythagorean theorem to determine if the given integer lengths can form a right triangle by assigning the largest number to \( c \) (hypotenuse), and the other two to \( a \) and \( b \) (the legs). The theorem holds true if the square of the hypotenuse equals the sum of the squares of the other two sides. Understanding this theorem is essential not only in geometry problems but also in various applications of mathematics such as physics, engineering, and computer science.

Conditional Statements in Java

In Java, conditional statements control the flow of execution based on certain conditions being met. The most common conditional statements are if, if-else, and switch. In the context of determining whether three sides make a right triangle, you'd use an if statement to compare the squared lengths of the sides.

Example of Conditional Statement:

If the condition within the if statement's parentheses evaluates to true, the block of code following it will execute. For instance:

\[ if (hypotenuse^2 == sideA^2 + sideB^2) { \]
System.out.println('The sides can form a right triangle.');
\[ } else { \]
System.out.println('The sides do not form a right triangle.');
\[ } \]

Through this structure, the program decides which message to display based on the validity of the Pythagorean theorem. The simplicity and versatility of conditional statements make them a vital part of programming logic in Java and other languages.

Input Validation

Input validation is a critical part of programming that involves ensuring user input matches the expected format, type, and value range. In Java, validating input prevents errors and ensures the application behaves as intended. For our right triangle exercise, we need to make sure that the user inputs non-zero integers, as zero or negative lengths do not make sense for triangle sides.

Example of Input Validation:

A simple validation could look like this:

\[ if (side1 > 0 && side2 > 0 && side3 > 0) { \]
// Proceed with the rest of the program.
\[ } else { \]
System.out.println('Please enter non-zero integers.');
\[ } \]

Implementing input validation prevents processing invalid or unexpected data, which can lead to wrong outputs or program crashes. As with any form of user interaction, robust input validation can greatly improve the reliability and user experience of software applications.