Question

Write a CGI script that receives as input three numbers from the client and displays a statement indicating whether the three numbers could represent an equilateral triangle (all three sides are the same length), an isosceles triangle (two sides are the same length) or a right triangle (the square of one side is equal to the sum of the squares of the other two sides).

Short answer

Check if all sides are equal for equilateral, two sides equal for isosceles, and use the Pythagorean theorem for a right triangle.

Step-by-step solution

Validate Input

First, ensure the CGI script receives three numerical inputs correctly. Use CGI parameters to fetch these inputs and validate that they are positive numbers, as a triangle cannot have zero or negative side lengths.

Check Equilateral Triangle Condition

Check if all three inputs are equal. If the numbers 'a', 'b', and 'c' satisfy the condition \( a = b = c \), the triangle is equilateral. Display an appropriate message to the client if this condition is true.

Check Isosceles Triangle Condition

Next, check if any two of the three sides are equal. This can be verified by checking if \( a = b \), \( b = c \), or \( a = c \). If any of these conditions hold true, the triangle is isosceles. Display the corresponding message.

Check Right Triangle Condition

For a right triangle, one must check the Pythagorean theorem. Verify if \( a^2 + b^2 = c^2 \), \( a^2 + c^2 = b^2 \), or \( b^2 + c^2 = a^2 \). If any of these equations hold true, the triangle is a right triangle. Display this message.

Determine and Display Triangle Type

If none of the above conditions are satisfied, determine that the triangle is scalene and not equilateral, isosceles, or right. Display that the provided sides represent a scalene triangle.

Key concepts

triangle type detection

When determining the type of triangle based on its sides, the process involves comparing the lengths to meet specific conditions. CGI scripting, a powerful tool for web-based programming, helps automate this detection by allowing us to dynamically process input values and execute scripts in response to HTTP requests.
For triangle type detection, we need to:

• Ensure the input consists of three positive numbers, as negative values or zeros cannot form a triangle.
• Analyze the side lengths to check for the specific conditions of equilateral, isosceles, or right triangles.
• Display the result to users, making the process straightforward and efficient.

Utilizing CGI in this context enriches interactive web applications by giving immediate feedback on the typed triangle. By processing input numbers, CGI scripts can efficiently determine the triangle type and present it to the user.

equilateral, isosceles, right triangles

Triangles can be classified into various types based on the side lengths and properties.

**Equilateral Triangles** are characterized by all three sides having the same length. This means that if the input values for the sides are such that \(a = b = c\), then the triangle is equilateral.

**Isosceles Triangles** have two sides that are equal in length. This is identified when any two sides, like \(a = b\), \(b = c\), or \(a = c\), are equal. The equality of these sides leads to two equal angles, giving the isosceles triangle a unique appearance.

**Right Triangles** follow the Pythagorean theorem, meaning the squares of the lengths must satisfy certain conditions. For a right triangle, if \(a, b, c\) represent the triangle's sides, then one of the expressions \(a^2 + b^2 = c^2\), \(a^2 + c^2 = b^2\), or \(b^2 + c^2 = a^2\) must hold true. This highlights the right-angle property of the triangle, often found in various applications like architecture and design.

Pythagorean theorem verification

The Pythagorean theorem is a fundamental principle for recognizing right triangles and plays a critical role in mathematics.

To verify if a given set of side lengths form a right triangle, each combination of the sides must be evaluated using the theorem. This theorem states that for any right triangle,

• the sum of the squares of the two shorter sides must equal the square of the largest side.

Mathematically, this is expressed as \(a^2 + b^2 = c^2\).

When using CGI scripting for this verification, you would check each possible permutation:

• Check \(a^2 + b^2 = c^2\)
• Check \(a^2 + c^2 = b^2\)
• Check \(b^2 + c^2 = a^2\)

Only one of these needs to be true for the triangle to be right-angled. This systematic check ensures that the Pythagorean condition is met, confirming the precise nature of the triangle. Use of algorithms or scripts aids in rapidly determining whether this condition is fulfilled for given side lengths.