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Converse of the Pythagorean Theorem — Definition, Formula & Examples

The Converse of the Pythagorean Theorem is a test for right triangles: if the squares of the two shorter sides of a triangle add up to the square of the longest side, then the triangle must be a right triangle.

Given a triangle with side lengths aa, bb, and cc where cc is the greatest, if a2+b2=c2a^2 + b^2 = c^2, then the triangle is a right triangle with the right angle opposite the side of length cc. This is the logical converse of the Pythagorean Theorem, which starts by assuming the triangle is already right-angled.

Key Formula

If a2+b2=c2, then the triangle is a right triangle.\text{If } a^2 + b^2 = c^2, \text{ then the triangle is a right triangle.}
Where:
  • aa = Length of one of the two shorter sides
  • bb = Length of the other shorter side
  • cc = Length of the longest side (potential hypotenuse)

How It Works

The original Pythagorean Theorem says "if a triangle is right, then a2+b2=c2a^2 + b^2 = c^2." The converse reverses the direction: "if a2+b2=c2a^2 + b^2 = c^2, then the triangle is right." To apply it, identify the longest side and call it cc. Compute a2+b2a^2 + b^2 and compare it to c2c^2. If they are equal, the triangle has a 90° angle opposite side cc. If a2+b2>c2a^2 + b^2 > c^2, the triangle is acute; if a2+b2<c2a^2 + b^2 < c^2, it is obtuse.

Worked Example

Problem: A triangle has side lengths 9, 12, and 15. Is it a right triangle?
Identify the longest side: The longest side is 15, so set c=15c = 15, a=9a = 9, and b=12b = 12.
Compute $a^2 + b^2$: Square the two shorter sides and add them.
92+122=81+144=2259^2 + 12^2 = 81 + 144 = 225
Compute $c^2$: Square the longest side.
152=22515^2 = 225
Compare: Since a2+b2=c2a^2 + b^2 = c^2, the converse tells us this is a right triangle with the right angle opposite the side of length 15.
Answer: Yes, the triangle with sides 9, 12, and 15 is a right triangle.

Why It Matters

Carpenters and builders use the 3-4-5 rule (a special case of this converse) to verify that corners are square. In geometry courses aligned to CCSS 8.G.B.6, proving the converse is a key exercise in understanding that a theorem and its converse are independent statements requiring separate proofs.

Common Mistakes

Mistake: Assuming the Pythagorean Theorem automatically works in both directions without a separate proof.
Correction: A theorem and its converse are logically distinct. The Pythagorean Theorem proves a2+b2=c2a^2 + b^2 = c^2 from a right angle; the converse proves a right angle from a2+b2=c2a^2 + b^2 = c^2. Each requires its own justification.

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