Solution**: Check if the Pythagorean theorem holds: - Capace Media
Solution: How to Check if the Pythagorean Theorem Holds
Solution: How to Check if the Pythagorean Theorem Holds
Coding or math enthusiasts know the power of the Pythagorean Theorem—a fundamental principle in geometry that applies to right-angled triangles. Whether you're a student learning the theorem or a programmer validating geometric relationships in an app, knowing how to check if the theorem holds is essential.
This article explores practical solutions to verify whether the Pythagorean Theorem a = b² + c² is true for any triangle, with a special focus on right triangles, and how you can automate this check using Python.
Understanding the Context
📐 What is the Pythagorean Theorem?
The Pythagorean Theorem states that in a right-angled 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 other two sides:
a² + b² = c²,
where c is the hypotenuse, and a and b are the other two sides.
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Key Insights
⚠️ Important: The theorem only holds for right-angled triangles. If the triangle isn’t right-angled, this equation will not hold.
✅ How to Check if the Pythagorean Theorem Holds
Here’s a step-by-step guide to determine whether a triangle satisfies the Pythagorean Theorem:
1. Identify the Triangle Type
Ensure the triangle has a right angle. This is crucial—otherwise, the theorem is not applicable.
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2. Measure the Side Lengths
Let a, b, and c be the lengths of the triangle’s sides. Identify the hypotenuse—this is the longest side.
3. Apply the Theorem
Check if: a² + b² = c²
a² + b² = c²or
a² + c² = b² or b² + c² = a²Depending on which side is the hypotenuse.
💻 Programmatically Check the Theorem Using Python
Automating this verification is useful for educational tools, geometry validation, or geometry-based games. Below is a simple and robust Python solution.
💡 Sample Python Code to Check the Pythagorean Theorem
# Test examples</code></pre><p>test_cases = [<br/> (3, 4, 5), # Right triangle<br/> (5, 12, 13), # Right triangle<br/> (1, 1, 1), # Not right-angled<br/> (0, 0, 0), # Degenerate case<br/> (2.5, 3.5, 4.5), # Approximate right triangle<br/>]
for a, b, c in test_cases:<br/> result = checks_pythagorean(a, b, c)<br/> print(f"Checking {a}, {b}, {c} → {'✅ Holds Theorem' if result else '❌ Does NOT hold'}")<br/><code>``
### 🔍 Explanation of the Code:- The function sorts the sides so the largest is assumed to be the hypotenuse.- It checks the equation with a small tolerance (</code>1e-9<code>) to account for floating-point precision issues.- The test cases include both valid right triangles and real-world approximations.