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

Check if the largest side squared equals the sum of the squares of the other two sides.

Step-by-step solution

Understand The Problem

We need to check if three given integers can be the sides of a right triangle. For a right triangle, the Pythagorean theorem should hold: the square of the largest side should be equal to the sum of the squares of the other two sides.

Identify Potential Hypotenuse

Given three integers, we first select the largest one as the potential hypotenuse and the other two as the potential legs of the triangle. This is crucial as the Pythagorean theorem (\( c^2 = a^2 + b^2 \)) requires that we identify which side is the hypotenuse (\( c \)).

Apply The Pythagorean Theorem

Check the condition of the Pythagorean theorem for a right triangle: \( c^2 = a^2 + b^2 \). If the condition is true, then the sides form a right triangle. Compute the squares of the sides and check the relation.

Implement Logic in Code

Implement a program that accepts three integer inputs. Compute the squares of these integers and check if the sum of squares of the two smaller integers equals the square of the largest integer. Use conditional statements to determine if the sides match the Pythagorean theorem.

Key concepts

Right Triangle

In geometry, a right triangle is a type of triangle where one of the angles measures exactly 90 degrees, or a right angle. This distinctive feature sets it apart from other types of triangles such as equilateral or isosceles triangles. The side opposite the right angle is known as the hypotenuse, which is the longest side in a right triangle.

The other two sides are called the legs of the triangle. In any right triangle, the Pythagorean theorem holds. The theorem provides a simple equation: the square of the length of the hypotenuse (\( c \)) is equal to the sum of the squares of the lengths of the other two sides (\( a \) and \( b \)). The formula is written as:

• \( c^2 = a^2 + b^2 \)

Understanding this equation is crucial when verifying if a set of three integers can serve as the side lengths for a right triangle. Recognizing the largest integer as the hypotenuse and other two as the legs establishes a foundation for applying the formula correctly.

Integer Sides

When considering integer sides for a triangle, particularly a right triangle, we focus on whole numbers. Using integers simplifies the calculation process but also demands accuracy to satisfy the conditions of the Pythagorean theorem.

To determine if three integers can form a right triangle, these steps must be followed:

• Identify the largest integer to use as the possible hypotenuse.
• Consider the other two integers as the potential legs.
• Apply the Pythagorean theorem to see if it holds true with these numbers.

If the theorem holds, then the sides form a right triangle. This approach relies on algebraic manipulation of integers to reveal relationships that satisfy geometric conditions.

Conditional Statements

Conditional statements are essential in programming when determining if three integers can form a right triangle. These statements allow a program to choose different paths of execution based on whether specific conditions are met.

In the context of determining right triangle side lengths, conditional statements check if the Pythagorean theorem is satisfied by the integers provided. The steps are:

• Accept three integer inputs from the user.
• Calculate the square of each integer.
• Use 'if-else' conditions to verify if the sum of the squares of the two smaller integers equals the square of the largest integer.

If the condition evaluates to true, the printed result is that the integers can represent the sides of a right triangle. Conditional statements thus serve as logic gates that either allow or block the path to this conclusion, based on math-driven verification.