Pythagorean Inequality Theorem: Acute and Obtuse Triangles

📐 Understanding the Pythagorean Inequality Theorem

The Pythagorean Inequality Theorem extends the well-known Pythagorean Theorem ($a^2 + b^2 = c^2$) to classify triangles as acute or obtuse based on the relationship between the squares of their side lengths. Here's how it works:

• Acute Triangle: If $a^2 + b^2 > c^2$, where $c$ is the longest side, then the triangle is acute (all angles are less than 90°).
• Obtuse Triangle: If $a^2 + b^2 < c^2$, where $c$ is the longest side, then the triangle is obtuse (one angle is greater than 90°).
• Right Triangle: If $a^2 + b^2 = c^2$, then the triangle is a right triangle (one angle is exactly 90°). This is the standard Pythagorean Theorem.

📝 Steps to Classify a Triangle

1. Identify the longest side: Let's call it 'c'.
2. Calculate $a^2 + b^2$: Where 'a' and 'b' are the shorter sides.
3. Calculate $c^2$: Square the longest side.
4. Compare: Determine if $a^2 + b^2 > c^2$, $a^2 + b^2 < c^2$, or $a^2 + b^2 = c^2$.
5. Classify: Based on the comparison, classify the triangle as acute, obtuse, or right.

✍️ Examples

Example 1: Acute Triangle

Sides: a = 5, b = 7, c = 8

Calculation:

• $a^2 + b^2 = 5^2 + 7^2 = 25 + 49 = 74$
• $c^2 = 8^2 = 64$

Comparison: $74 > 64$. Therefore, this is an acute triangle.

Example 2: Obtuse Triangle

Sides: a = 6, b = 8, c = 11

Calculation:

• $a^2 + b^2 = 6^2 + 8^2 = 36 + 64 = 100$
• $c^2 = 11^2 = 121$

Comparison: $100 < 121$. Therefore, this is an obtuse triangle.

Example 3: Right Triangle

Sides: a = 3, b = 4, c = 5

Calculation:

• $a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25$
• $c^2 = 5^2 = 25$

Comparison: $25 = 25$. Therefore, this is a right triangle.

💻 Code Example (Python)

def triangle_type(a, b, c):
 a, b, c = sorted([a, b, c]) # Sort sides to ensure c is the longest
 if a**2 + b**2 > c**2:
 return "Acute"
 elif a**2 + b**2 < c**2:
 return "Obtuse"
 else:
 return "Right"

print(triangle_type(5, 7, 8))   # Output: Acute
print(triangle_type(6, 8, 11))  # Output: Obtuse
print(triangle_type(3, 4, 5))   # Output: Right