Pythagoras Theorem is part of the Class 9 Mathematics syllabus for the AISSEE (Sainik School) entrance exam, where Maths carries 200 of 400 marks (50 questions × 4 marks), and for the RMS CET Class 9 paper. AISSEE 2027 is on Sunday, 31 January 2027, with no negative marking.
The Theorem
In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle, and the longest side) equals the sum of the squares of the other two sides: c² = a² + b².
Converse
If, in a triangle, the square of the longest side equals the sum of the squares of the other two sides, the triangle is right-angled. Example: 6, 8, 10 → 36 + 64 = 100 ✔ right-angled. 5, 6, 7 → 25 + 36 = 61 ≠ 49 ✘ not right-angled.
Pythagorean Triplets to Remember
Triplet | Multiples |
|---|---|
3, 4, 5 | 6, 8, 10 · 9, 12, 15 · 12, 16, 20 |
5, 12, 13 | 10, 24, 26 |
8, 15, 17 | 16, 30, 34 |
7, 24, 25 | 14, 48, 50 |
20, 21, 29 | — |
9, 40, 41 | — |
Knowing these saves time: if two sides match a triplet, write the third side straight away.
Solved Examples
Ladder: a 13 m ladder reaches a window; its foot is 5 m from the wall. Height = √(13² − 5²) = √144 = 12 m.
Rectangle diagonal: 8 cm × 6 cm. Diagonal = √(64 + 36) = 10 cm.
Rhombus: diagonals 16 cm and 12 cm bisect each other at right angles, so side = √(8² + 6²) = 10 cm.
Distance: walk 9 km north, then 12 km east. Distance from start = √(81 + 144) = 15 km.
Square: side 5 cm. Diagonal = 5√2 ≈ 7.07 cm.
Exam Tips
The hypotenuse is always the longest side — check this before substituting.
Learn squares from 1 to 30 for quick square roots.
In direction-sense and ladder questions, draw a quick sketch first.
Practise and Revise
Pythagoras Theorem MCQs — 10 questions with answers and explanations
Free AISSEE / RMS Mock Test
Enunciate Academy runs online coaching for Sainik School, RMS, Navodaya and RIMC entrance exams for Class 4 to 9 students across Andhra Pradesh and Telangana. Message us on WhatsApp (94924 44498) to get a free mock test, or see the Spark batch details.
