What is a converse?
The converse of a theorem swaps the "if" and the "then". The Pythagoras theorem says: if a triangle is right-angled, then c² = a² + b². The converse turns this around to give us a test for a right angle.
Key idea
Converse of Pythagoras theorem: if the three sides of a triangle satisfy c² = a² + b² (where c is the longest side), then the triangle is a right-angled triangle, with the right angle opposite the longest side.
How to test a triangle
Follow three steps:
- Identify the longest side and call it c.
- Work out c² and also a² + b² for the two shorter sides.
- If the two results are equal, the triangle has a right angle. If not, it does not.
Worked example
Is a triangle with sides 8 cm, 15 cm and 17 cm right-angled?
Longest side c = 17: c² = 17² = 289.
a² + b² = 8² + 15² = 64 + 225 = 289.
Since 289 = 289, the triangle is right-angled.
Worked example
Test sides 4 cm, 5 cm and 6 cm.
c² = 6² = 36; a² + b² = 4² + 5² = 16 + 25 = 41.
Since 36 ≠ 41, the triangle is not right-angled.
Remember
- Always use the longest side as c.
- Equal (c² = a² + b²) means a right angle exists.
- Not equal means no right angle.
- The right angle is opposite the longest side.