The Converse of the Pythagorean Theorem: Proving Right, Acute & Obtuse Triangles

The classical Pythagorean Theorem says:

If a triangle is a right triangle, then a² + b² = c².

The Converse of the Pythagorean Theorem asks the reverse question:

If three side lengths satisfy a² + b² = c², does the triangle have to be a right triangle?

The answer is yes. Furthermore, comparing the sum of the squares against the square of the longest side allows us to classify every possible triangle as right, acute, or obtuse.

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The Classification Rules

Let a, b, and c be the lengths of the three sides of a triangle, sorted so that c is the longest side (a ≤ b ≤ c).

First, verify that the sides satisfy the Triangle Inequality Theorem:

a + b > c

(If a + b ≤ c, the segments cannot meet to form any triangle at all).

Once confirmed, compute a² + b² and compare it directly to c²:

Condition	Triangle Classification	Angle Opposite Side c
a² + b² = c²	Right Triangle	Exactly 90°
a² + b² > c²	Acute Triangle	Strictly less than 90° (all angles acute)
a² + b² < c²	Obtuse Triangle	Strictly greater than 90° (one obtuse angle)

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Why Does This Work? (The Law of Cosines Intuition)

The algebraic reason behind this classification comes from the Law of Cosines, which applies to all triangles:

c² = a² + b² - 2ab · cos(θ)

where θ is the angle opposite side c:

1. If θ = 90°, then cos(90°) = 0, reducing to c² = a² + b².
2. If θ < 90° (acute), then cos(θ) > 0. Subtracting a positive number makes c² < a² + b², or a² + b² > c².
3. If θ > 90° (obtuse), then cos(θ) < 0. Subtracting a negative number increases c², so c² > a² + b², or a² + b² < c².

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Worked Examples

Example 1: Sides 6, 8, 10

• Longest side: c = 10
• Triangle inequality: 6 + 8 = 14 > 10 (Valid)
• Sum of squares: 6² + 8² = 36 + 64 = 100
• c² = 10² = 100
• Since 100 = 100, this is a Right Triangle.

Example 2: Sides 5, 7, 8

• Longest side: c = 8
• Triangle inequality: 5 + 7 = 12 > 8 (Valid)
• Sum of squares: 5² + 7² = 25 + 49 = 74
• c² = 8² = 64
• Since 74 > 64, this is an Acute Triangle.

Example 3: Sides 4, 7, 9

• Longest side: c = 9
• Triangle inequality: 4 + 7 = 11 > 9 (Valid)
• Sum of squares: 4² + 7² = 16 + 49 = 65
• c² = 9² = 81
• Since 65 < 81, this is an Obtuse Triangle.

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