The theorem
In any right-angled triangle, the square of the longest side (the hypotenuse, opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c².
It only works when the triangle has a 90° angle. Identify the hypotenuse first, every single time — it is always opposite the right angle and always the longest side.
Finding the hypotenuse
Legs of 3 and 4: c² = 3² + 4² = 9 + 16 = 25, so c = 5.
Legs of 6 and 8: c² = 36 + 64 = 100, so c = 10. Legs of 5 and 12 give c = 13. These three sets (3-4-5, 6-8-10, 5-12-13) show up constantly — recognising them saves time.
Finding a missing leg
Here you subtract instead of add. If the hypotenuse is 13 and one leg is 5: 5² + b² = 13², so 25 + b² = 169, b² = 144, b = 12.
The most common error in the whole topic is adding when you should subtract. Ask yourself: is the side I am looking for the longest one? If not, subtract.
Checking if a triangle is right-angled
Square all three sides. If the two smaller squares add up to the largest square, the triangle is right-angled. Sides 9, 12, 15: 81 + 144 = 225 = 15². Yes. Sides 4, 5, 8: 16 + 25 = 41, but 8² = 64. No.
Word problems
A ladder 10 m long leans against a wall with its foot 6 m from the base. How high does it reach? The ladder is the hypotenuse: 6² + h² = 10², so h² = 100 - 36 = 64 and h = 8 m.
Draw the triangle before touching the algebra. Almost every ladder, ramp, diagonal-of-a-rectangle or distance-between-two-points question is this theorem wearing a costume.