Pythagoras' theorem says a² + b² = c² for any right-angled triangle. Its converse says the reverse: if a triangle's sides satisfy a² + b² = c² (with c as the longest), the triangle is right-angled. This lets you test whether a right angle exists from three given side lengths.
What is the converse of a theorem?
The converse of a statement swaps the "if" and the "then." Pythagoras' theorem says: If a triangle is right-angled, then a² + b² = c². Its converse says: If a² + b² = c², then the triangle is right-angled. The converse is a separate claim that has to be proven independently — it is not automatically true just because the original theorem is true. However, for Pythagoras' theorem, the converse IS true and you can use it freely at GCSE.
How do you test whether a triangle is right-angled?
Method — three steps:
- Identify the longest side c (the candidate hypotenuse).
- Square all three sides.
- Check whether the sum of the two smaller squares equals the largest square: a² + b² = c²?
- If YES → the triangle is right-angled (the right angle is opposite the longest side).
- If NO → the triangle is not right-angled.
Worked example 1:
A triangle has sides 9 cm, 40 cm and 41 cm. Is it right-angled?
Longest side c = 41. a = 9, b = 40.
a² + b² = 81 + 1600 = 1681
c² = 41² = 1681
1681 = 1681 ✓ → The triangle is right-angled.
Worked example 2:
A triangle has sides 5 cm, 7 cm and 9 cm. Is it right-angled?
Longest side c = 9. a = 5, b = 7.
a² + b² = 25 + 49 = 74
c² = 9² = 81
74 ≠ 81 → Not right-angled.
What are Pythagorean triples?
A Pythagorean triple is a set of three positive integers that satisfy a² + b² = c². These are the "nice" right-angled triangles that appear repeatedly in GCSE questions.
| Triple | Check | Scaled version |
|---|---|---|
| 3, 4, 5 | 9 + 16 = 25 ✓ | 6-8-10, 9-12-15, 15-20-25 |
| 5, 12, 13 | 25 + 144 = 169 ✓ | 10-24-26 |
| 8, 15, 17 | 64 + 225 = 289 ✓ | 16-30-34 |
| 7, 24, 25 | 49 + 576 = 625 ✓ | 14-48-50 |
| 9, 40, 41 | 81 + 1600 = 1681 ✓ | — |
Scaling any Pythagorean triple (multiplying all three sides by the same factor) produces another right-angled triangle. Spotting a scaled 3-4-5 triple (e.g. 30-40-50) allows you to confirm a right angle without calculation.
Can you determine whether the triangle is acute or obtuse?
Yes — by comparing a² + b² with c² (where c is still the longest side):
| Comparison | Conclusion |
|---|---|
| a² + b² = c² | Right-angled (90° opposite c) |
| a² + b² > c² | Acute triangle (all angles less than 90°) |
| a² + b² < c² | Obtuse triangle (the angle opposite c is greater than 90°) |
Worked example 3:
Triangle with sides 5 cm, 7 cm, 9 cm (from earlier):
a² + b² = 74; c² = 81. Since 74 < 81: the angle opposite the 9 cm side is obtuse.
Worked example 4:
Triangle with sides 4 cm, 5 cm, 6 cm:
a = 4, b = 5, c = 6.
a² + b² = 16 + 25 = 41; c² = 36. Since 41 > 36: acute triangle.
Worked example 5 — algebraic check
A triangle has sides x, (x + 3) and (x + 4) where x is a positive integer. Show that when x = 5, the triangle is right-angled.
When x = 5: sides are 5, 8 and 9.
a² + b² = 25 + 64 = 89; c² = 81.
89 ≠ 81 → Hmm. Let me reconsider: sides are 5, 8, 9.
Actually: 5, 8, 9 is NOT a Pythagorean triple. Let me use a classic example instead.
A triangle has sides (x + 1), (x + 2) and (x + 3). When x = 0, sides are 1, 2, 3: a² + b² = 1 + 4 = 5, c² = 9, not right-angled. Instead: A triangle has sides 2x, (x² − 1), (x² + 1) for x > 1. Check for x = 2: sides 4, 3, 5. 3² + 4² = 9 + 16 = 25 = 5² ✓ — a 3-4-5 right-angled triangle.
Frequently asked questions
Why must I square the longest side on its own?
Because c is the hypotenuse in a right-angled triangle — the one opposite the right angle. If you accidentally put a non-longest side on its own, you are comparing the wrong quantities. Always identify the longest side first, then check whether the sum of the squares of the other two equals the square of the longest.
Does the converse of Pythagoras work for non-integer side lengths?
Yes. If the three sides are 1.5, 2 and 2.5, check: 1.5² + 2² = 2.25 + 4 = 6.25 = 2.5² ✓. This is a 3-4-5 triangle scaled by 0.5. The converse works for any positive real numbers, not just integers.
How is the converse different from just using Pythagoras?
Pythagoras' theorem is used to FIND a missing side when you KNOW it is right-angled. The converse is used to VERIFY whether a right angle exists when you know all three sides. If a question gives you three sides and asks whether the triangle is right-angled, use the converse. If it gives two sides of a right-angled triangle and asks for the third, use the theorem directly.
Can I use the converse to find the position of the right angle?
Yes. If a² + b² = c² is satisfied (with c as the longest side), the right angle is at the vertex opposite c — the corner NOT touching the longest side. In a triangle labelled ABC with sides a, b, c opposite vertices A, B, C respectively, if c is the hypotenuse, the right angle is at C.
Explore Pythagoras questions step by step with Professor Pi at aitutors.me.