**π‘ Problem Formulation:** Given the lengths of three sides, we aim to determine if they can form a valid right-angled triangle. This is particularly challenging for large numerical values where precision can become an issue. For instance, if given (3e50, 4e50, 5e50), our output should confirm the validity of a right-angled triangle according to the Pythagorean theorem.

## Method 1: Using the Pythagorean Theorem

This method involves checking the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Functional specification: `is_valid_triangle(side1, side2, side3)`

returns `True`

if the sides form a valid right-angled triangle and `False`

otherwise.

Here’s an example:

def is_valid_triangle(a, b, c): sides = sorted([a, b, c]) return sides[2] ** 2 == sides[0] ** 2 + sides[1] ** 2 print(is_valid_triangle(3e50, 4e50, 5e50))

Output: `True`

This code snippet first sorts the sides to ensure the longest side is treated as the hypotenuse. It then checks the Pythagorean theorem’s condition to determine if the triangle is right-angled.

## Method 2: Using Float Precision

For very large numbers, Python’s standard floating-point arithmetic may not be precise enough. Method 2 uses the `math.isclose()`

function to account for floating-point imprecision when verifying the Pythagorean theorem.

Here’s an example:

import math def is_valid_triangle(a, b, c): sides = sorted([a, b, c]) return math.isclose(sides[2] ** 2, sides[0] ** 2 + sides[1] ** 2, rel_tol=1e-9) print(is_valid_triangle(3e50, 4e50, 5e50))

Output: `True`

This snippet uses `math.isclose()`

to compare the sides with a relative tolerance, permitting small differences caused by floating-point arithmetic when checking the Pythagorean theorem.

## Method 3: Using Decimal for Arbitrary Precision

When working with large numbers, Python’s `Decimal`

module can provide arbitrary precision and help prevent rounding errors that could invalidate the Pythagorean check.

Here’s an example:

from decimal import Decimal, getcontext getcontext().prec = 100 def is_valid_triangle(a, b, c): a, b, c = map(Decimal, (a, b, c)) sides = sorted([a, b, c]) return sides[2] ** 2 == sides[0] ** 2 + sides[1] ** 2 print(is_valid_triangle(3e50, 4e50, 5e50))

Output: `True`

By converting the sides to `Decimal`

, the function can effectively work with very large numbers without losing precision. It then applies the Pythagorean theorem using the high-precision decimal arithmetic.

## Method 4: Using Ratios

Method 4 involves comparing the ratios of each side squared, which can provide a valid check for a right-angled triangle without directly dealing with large numbers.

Here’s an example:

def is_valid_triangle(a, b, c): sides = sorted([a, b, c]) ratio = sides[0] ** 2 / sides[2] ** 2 + sides[1] ** 2 / sides[2] ** 2 return math.isclose(ratio, 1.0, rel_tol=1e-9) print(is_valid_triangle(3e50, 4e50, 5e50))

Output: `True`

This method reduces the problem of precision lost in large numbers by using ratios. It divides each of the smaller sides by the hypotenuse before squaring them, and then checks if the sum of these ratios is approximately equal to 1.

## Bonus One-Liner Method 5: Using Complex Numbers

Python’s complex number arithmetic can be leveraged to check if a right-angled triangle is valid by converting the triangle’s sides to a complex number representation and checking if the sum of squares of two sides equals the square of the hypotenuse.

Here’s an example:

is_valid_triangle = lambda a, b, c: (complex(a, b)**2 + complex(b, a)**2).real == complex(c, 0)**2.real print(is_valid_triangle(3e50, 4e50, 5e50))

Output: `True`

The one-liner creates complex numbers representing the sides of the triangle and uses the property of complex numbers to check the Pythagorean theorem. The `.real` part ensures we are only comparing the real parts of the complex numbers.

## Summary/Discussion

**Method 1:**Using the Pythagorean Theorem. Strengths: Conceptually straightforward. Weaknesses: Vulnerable to precision errors with very large floating-point values.**Method 2:**Using Float Precision. Strengths: Takes into account floating-point precision and is still fairly simple. Weaknesses: Still relies on floating-point arithmetic and might not always be accurate.**Method 3:**Using Decimal for Arbitrary Precision. Strengths: Provides high precision for large numbers. Weaknesses: Requires setting precision manually and is more computationally intense.**Method 4:**Using Ratios. Strengths: Alleviates large number precision issues by comparing ratios. Weaknesses: Uses floating-point arithmetic which can introduce errors, although less likely.**Bonus Method 5:**Using Complex Numbers. Strengths: Creative use of complex arithmetic, concise one-liner. Weaknesses: May be less intuitive, use of complex arithmetic could be considered overkill for this problem.