Pythagoras’ theorem
Finding a length in a right-angled triangle, and above all understanding why it’s the squares that add up.
The idea in one sentence
In a right-angled triangle, the square built on the longest side (the hypotenuse) has the same area as the two squares built on the other two sides, put together.
BC² = AB² + AC²
We add areas, so squares of lengths. Not lengths. That’s where almost every mistake happens.
One complete reasoning
ABC is right-angled at A. AB = 6 cm, AC = 8 cm. How long is BC?
- Find the right angle: it’s at A. The hypotenuse is the opposite side, BC.BC = ?
- Write the relation on the squares.BC² = AB² + AC²
- Substitute and work out the squares.BC² = 36 + 64 = 100
- Take the square root, last of all.BC = 10 cm
The classic mistake
- Modelling
BC = 14 cm
The lengths were added. Tempting, because “they add up”. But it’s the areas of the squares that add.
- Execution
BC = 100 cm
The relation is right, 100 is right. But 100 is BC². The square root is missing.
- Reasoning
AB² = BC² + AC²
The hypotenuse was misidentified. It is always opposite the right angle, and always alone on one side of the equation.
Two exercises
DEF is right-angled at E. DE = 5 cm, EF = 12 cm. Find DF.Show answer
DF² = DE² + EF² = 25 + 144 = 169, so DF = 13 cm.
A triangle has sides 7 cm, 24 cm and 25 cm. Is it right-angled?Show answer
25² = 625 and 7² + 24² = 49 + 576 = 625. The equality holds: by the converse of Pythagoras, the triangle is right-angled, with the right angle opposite the 25 cm side.