The Baudhayana Sulbasutra stated this rule around 800 BCE, long before Pythagoras. In an Olympiad it is almost never asked directly — it is hidden inside a rectangle, a rhombus or a journey.
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a² + b² = c², where c is the hypotenuse — the side opposite the right angle, and always the longest.
📐 The triples to recognise instantly
| a | b | c | Check |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 |
| 5 | 12 | 13 | 25 + 144 = 169 |
| 6 | 8 | 10 | 36 + 64 = 100 |
| 7 | 24 | 25 | 49 + 576 = 625 |
| 8 | 15 | 17 | 64 + 225 = 289 |
| 9 | 12 | 15 | 81 + 144 = 225 |
Scaled triples are triples too. From 3-4-5 come 6-8-10, 9-12-15, 15-20-25 and 30-40-50. If the numbers in a question are a familiar triple times k, write the answer straight down.
The converse is examinable. If a² + b² = c² then the triangle is right-angled. So 8, 15, 17 is a right triangle because 64 + 225 = 289. But 5, 6, 7 is not, because 25 + 36 = 61 ≠ 49.
Half the marks in this topic come from spotting the triangle nobody drew for you.
🔍 Where they hide
A rectangle’s diagonal splits it into two right triangles. A 9 × 12 rectangle has a diagonal of 15.
A square of side a has diagonal a√2 — so a 10 cm square has a diagonal of about 14.1 cm.
A rhombus. Its diagonals bisect each other at right angles, so half of each diagonal makes a right triangle with the side. Diagonals 16 and 30 give a side of √(8² + 15²) = 17.
An equilateral triangle of side a has height a√3 ÷ 2.
A journey. "5 km north then 12 km east" is a right triangle: the straight-line distance is 13 km.
The ladder question, every year. A 25 m ladder rests with its foot 7 m from a wall. It reaches √(625 − 49) = √576 = 24 m up.
Area, perimeter and distance questions built on a right triangle.
Area of a right triangle = half the product of the two shorter sides. For 6, 8, 10 the area is ½ × 6 × 8 = 24 square units — the hypotenuse plays no part.
Perimeter first, then Pythagoras. "A right triangle has legs 9 and 12. Find its perimeter." The hypotenuse is 15, so the perimeter is 9 + 12 + 15 = 36.
The longest side is always the hypotenuse. If a question gives you 15 and 17 and asks for the third side, 17 is the hypotenuse, so the answer is √(289 − 225) = 8, not √514.
12 Olympiad questions on right triangles. Find the hidden triangle first. 🍀
