Pythagoras

In a right triangle the square on the hypotenuse equals the sum of the squares on the other two sides: a² + b² = c², where c is the hypotenuse. The fastest CAT students never crunch the algebra — they recognise the standard Pythagorean triples on sight: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and 9-40-41, together with every multiple (6-8-10, 9-12-15, 10-24-26, and so on). Spotting that 39 and 52 are 3 × 13 and 4 × 13 instantly gives a hypotenuse of 65. The converse is equally useful: if a² + b² = c² the triangle is right-angled, while a² + b² > c² makes it acute and a² + b² < c² obtuse, which lets you classify a triangle from its sides alone. Two special right triangles recur constantly: the 45-45-90 with sides in ratio 1 : 1 : √2, and the 30-60-90 with sides in ratio 1 : √3 : 2 — memorise both, because hexagons, equilateral triangles and squares cut along a diagonal all produce them.

✅ Solved examples

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📝 Topic test — 8 questions

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