The Converse of the Pythagorean Theorem

The Pythagorean theorem tells us how to compute the third side of a right triangle from the other two. The converse of the Pythagorean theorem goes the other direction: from three sides it helps us decide whether a triangle is a right triangle at all.

---

Table of contents

• Statement of the converse
• How to check
• Solved examples
• The most famous right triangle triples
• Common mistakes

---

Statement of the converse

> Converse of the Pythagorean theorem: If in a triangle with sides $a$, $b$, $c$ (where $c$ is the longest side) the equation

>

>

>

> holds, then the triangle is a right triangle and the right angle lies opposite side $c$.

The important thing is that both implications hold:

• If the triangle is right → the relation $a^2+b^2=c^2$ holds.
• If the relation $a^2+b^2=c^2$ holds → the triangle is right.

---

How to check

1. Find the longest side. Label it $c$ – if the triangle is right, this will be the hypotenuse.
2. Compute $a^2+b^2$ from the other two sides.
3. Compute $c^2$ from the longest side.
4. Compare the results:

- If $a^2+b^2=c^2$, the triangle is a right triangle.

- If $a^2+b^2\ne c^2$, the triangle is not a right triangle.

---

Solved examples

Example 1: $a=6$, $b=8$, $c=10$.

The longest side is $10$.

$$a^2+b^2=6^2+8^2=36+64=100$$
$$c^2=10^2=100$$
$$100=100✓$$

The triangle is a right triangle.

---

Example 2: $a=5$, $b=6$, $c=8$.

The longest side is $8$.

$$5^2+6^2=25+36=61$$
$$8^2=64$$
$$61\ne 64$$

The triangle is not a right triangle.

---

Example 3: $a=9$, $b=40$, $c=41$.

$$9^2+40^2=81+1600=1681$$
$$41^2=1681$$
$$1681=1681✓$$

Yes, it is a right triangle.

---

The most famous right triangle triples

Some triples of integers show up often in problems – it's worth learning them by heart:

• $(3,4,5)$
• $(5,12,13)$
• $(6,8,10)$ – a multiple of $(3,4,5)$
• $(7,24,25)$
• $(8,15,17)$
• $(9,40,41)$
• $(20,21,29)$

> 👉 More information: Pythagorean triples

---

Common mistakes

⚠️ Don't forget to check which side is the longest. If you accidentally plug a different side in for $c$, the result won't be correct.

⚠️ You compare the squares, not the sides. It often happens that a student just adds the sides instead of their squares.

⚠️ Watch out for rounding accuracy. If the sides aren't whole numbers, even a small measurement error can make the equality not exactly hold. In that case it's enough if the difference is negligible.

---

Related articles

• The Pythagorean theorem – complete guide
• The Pythagorean theorem formula
• Pythagorean theorem examples
• Pythagorean triples

Practice

• ✏️ Checking for a right triangle
• 🎯 Mixed problems