Class 8 Maths · Chapter 9
Short answer: Chapter 9 "The Baudhayana-Pythagoras Theorem" naye NCERT Ganita Prakash Class 8, Part 2 ka geometry chapter hai — right triangles ki teeno sides ke beech ka relationship (Baudhayana-Pythagoras theorem) sikhata hai, proof intuition, unknown side nikalna, aur real-life applications (ladder, diagonal, distance problems) cover karta hai.
Class 8 ki Maths book is session (2026-27) se completely naye format me aa gayi hai — purani single "Mathematics" book ki jagah ab NCERT ne Ganita Prakash naam se do-part edition launch kiya hai (Part 1: Chapters 1-7, Part 2: Chapters 8-14), total 14 chapters. Bahut se students aur parents ko pehle to "class 8 maths new book name 2026-27" search karke hi pata chalta hai ki naam badal gaya hai. Ye chapter — "The Baudhayana-Pythagoras Theorem" — Part 2 ka pehla chapter (Chapter 9) hai, aur geometry ke us sabse famous result ko cover karta hai jo aap shayad "Pythagoras theorem" naam se pehle se jaante honge, lekin NCERT ne ise deliberately Baudhayana-Pythagoras Theorem naam diya hai — kyunki ye result Baudhayana ke Sulba Sutra (lagbhag 800 BCE, Pythagoras se sadiyon pehle) me already likha ja chuka tha.
Ye chapter right-angled triangles ki teeno sides — do "legs" (perpendicular aur base) aur ek "hypotenuse" (sabse lambi side, right angle ke saamne) — ke beech ek fixed numerical relationship establish karta hai: hypotenuse ka square = doosri do sides ke squares ka sum. Chapter me is relationship ko sirf ratt-ne ke bajaye, square-grid diagrams aur area-based visual proofs se samajhne par zor diya gaya hai — jo Ganita Prakash ki overall philosophy (conceptual understanding pehle, formula baad me) se match karta hai.
Poore Ganita Prakash Part 1 aur Part 2 ka chapter list dekhein to pattern clear hai: Part 1 me "A Square and A Cube" (numbers), "Proportional Reasoning 1 vs 2", "We Distribute Yet Things Multiply" (algebra distributive law), "Fractions in Disguise", "Number Play", "Tales by Dots and Lines" (graphs), "Algebra Play" jaise chapters hain — sab naye, thematic naam waale. Part 2 me is Baudhayana-Pythagoras chapter ke saath "Exploring Some Geometric Themes", "Area" aur "Quadrilaterals" jaise geometry-heavy chapters hain, jo dikhata hai ki purani book ka pura geometry section rationalise ho kar naye tarike se present kiya gaya hai.
Is spec me hum theorem ka intuition, standard exercise-style solved examples, common mistakes, aur exam-relevant practice cover karenge — taaki aap chapter conceptually bhi samjhein aur exercise questions bhi confidently solve kar sakein.
Chapter 9 Summary — 5 Minute Revision
"The Baudhayana-Pythagoras Theorem" Class 8 Ganita Prakash Part 2 ka geometry chapter hai jo right-angled triangles ki teeno sides ke beech fundamental numerical relationship establish karta hai. Chapter ka core idea simple hai lekin powerful hai: agar ek triangle me ek angle exactly 90° ka hai, to us right angle ke saamne waali side (jise hypotenuse kehte hain) ka square, baaki do sides ke squares ke sum ke barabar hota hai.
(Hypotenuse)2 = (Base)2 + (Height/Perpendicular)2
Chapter historical context se shuru hota hai — Baudhayana ke Sulba Sutra (Vedic-era text jo altar/yagna construction ke liye geometric rules deta tha) me ye theorem Pythagoras se pehle hi likha gaya tha, isliye NCERT ne isko "Baudhayana-Pythagoras Theorem" naam diya hai, sirf "Pythagoras Theorem" nahi. Ye naming choice deliberate hai aur exam me bhi is naam se hi refer kiya jaana chahiye.
Theorem ko chapter me graph-paper/square-grid activities ke through visually derive kiya gaya hai — teeno sides par squares banakar unke areas count karke dikhaya jaata hai ki bade square (hypotenuse waale) ka area = do chhote squares ka combined area. Ye approach students ko formula ratt-ne ke bajaye area conservation ke through samajhne me madad karti hai.
Chapter me phir is relationship ko use karke practical problems solve karwaye jaate hain: kisi bhi ek missing side ko doosri do sides se nikalna, real-life situations (ladder wall se kitni door hai, diagonal distance, rectangle ki diagonal) me apply karna, aur Pythagorean triplets (jaise 3-4-5, 5-12-13) identify karna jo bina calculator ke turant answer de dete hain.
Yaad rakhne waali sabse important baat: ye theorem sirf right-angled triangles par lagu hota hai — agar triangle me 90° ka angle nahi hai to ye relationship apply nahi hogi. Aur hypotenuse hamesha right angle ke saamne waali (sabse lambi) side hoti hai, kisi bhi random side ko hypotenuse maan lena sabse common mistake hai jo students karte hain.
In-Text Questions — Solutions
Ex 1: Ek right-angled triangle me base = 6 cm aur height = 8 cm hai. Hypotenuse ki length nikalo.
Baudhayana-Pythagoras theorem se:
(Hypotenuse)2 = (Base)2 + (Height)2
(Hypotenuse)2 = 62 + 82 = 36 + 64 = 100
Hypotenuse = √100 = 10 cm
Isliye hypotenuse ki length 10 cm hai. (Ye 3-4-5 triplet ka scaled version hai: 6-8-10 = 2 × 3-4-5.)
Ex 2: Ek right-angled triangle ka hypotenuse 13 cm hai aur ek side 5 cm hai. Doosri side nikalo.
Theorem ko rearrange karke:
(Doosri side)2 = (Hypotenuse)2 − (Di gayi side)2
(Doosri side)2 = 132 − 52 = 169 − 25 = 144
Doosri side = √144 = 12 cm
Isliye missing side 12 cm hai — ye 5-12-13 ka Pythagorean triplet hai.
Ex 3: Ek 10 m lambi ladder ek deewar se 6 m door ground par rakhi hai. Ladder deewar par kitni height tak pahunchegi?
Ladder, deewar, aur ground ek right-angled triangle banate hain jahan ladder hypotenuse hai (10 m), aur deewar se ground distance ek side hai (6 m).
(Height)2 = (Ladder)2 − (Ground distance)2 = 102 − 62 = 100 − 36 = 64
Height = √64 = 8 m
Ladder deewar par 8 m ki height tak pahunchegi.
Exercise Questions — Solutions (Q1–Q5)
Ek right-angled triangle ke do legs (perpendicular sides) 9 cm aur 12 cm hain. Hypotenuse nikalo.
(Hypotenuse)2 = 92 + 122 = 81 + 144 = 225
Hypotenuse = √225 = 15 cm
Ek rectangle ki length 24 cm aur breadth 7 cm hai. Iski diagonal ki length nikalo.
Rectangle ki diagonal, length aur breadth ke saath ek right-angled triangle banati hai (diagonal = hypotenuse).
(Diagonal)2 = 242 + 72 = 576 + 49 = 625
Diagonal = √625 = 25 cm
Check karo ki 8 cm, 15 cm, aur 17 cm sides waala triangle right-angled hai ya nahi.
Sabse badi side (17 cm) ko hypotenuse maan kar check karte hain:
172 = 289
82 + 152 = 64 + 225 = 289
Dono sides equal hain (289 = 289), isliye ye right-angled triangle hai, aur right angle 17 cm side ke saamne hai.
Do khambe (poles) ek doosre se 12 m door khade hain. Unki heights 10 m aur 15 m hain. Unke tops ke beech ki distance nikalo.
Heights ka difference = 15 − 10 = 5 m. Ye vertical side hai, aur khambon ke beech horizontal distance 12 m hai. Tops ke beech ki distance hypotenuse hogi.
(Distance)2 = 122 + 52 = 144 + 25 = 169
Distance = √169 = 13 m
Ek square ka side 14 cm hai. Uski diagonal nikalo (√2 ≈ 1.414 use karo).
Square ki diagonal, do sides ke saath right-angled triangle banati hai jisme dono legs 14 cm equal hain.
(Diagonal)2 = 142 + 142 = 196 + 196 = 392
Diagonal = √392 = 14√2 ≈ 14 × 1.414 = 19.8 cm (approx)
Important Equations — Ek Nazar Me
| Rule / Concept | Formula |
|---|---|
| Baudhayana-Pythagoras Theorem | (Hypotenuse)2 = (Base)2 + (Height)2 |
| Missing leg nikalna | (Leg)2 = (Hypotenuse)2 − (Doosri leg)2 |
| Rectangle ki diagonal | Diagonal2 = Length2 + Breadth2 |
| Square ki diagonal | Diagonal = Side × √2 |
| Common Pythagorean triplets | 3-4-5, 5-12-13, 8-15-17, 7-24-25 (aur inke multiples jaise 6-8-10) |
| Right-angle check condition | Agar sabse badi side2 = doosri do sides ke squares ka sum, tabhi triangle right-angled hai |
↔ Table ko side me swipe karein
Common Mistakes — Yahan Marks Kat te Hain
- Kisi bhi random side ko hypotenuse maan lena — hypotenuse hamesha right angle ke saamne waali aur sabse lambi side hoti hai, ise pehchaan kar hi formula lagao.
- Theorem ko non-right-angled triangles par apply karne ki koshish karna — ye relationship sirf right-angled triangles ke liye valid hai.
- Squares add karne ke bajaye directly sides add kar dena (jaise 6 + 8 = 14 likh dena, jabki sahi answer √(6² + 8²) = 10 hai).
- Missing leg nikalte waqt subtraction ka order galat kar dena — hamesha (Hypotenuse)² − (Di gayi leg)² hoga, kabhi ulta nahi.
- Square root nikalte waqt calculation error — final step me √ lagana bhoolna aur square value ko hi final answer bata dena.
- Units ko square karna bhool jaana ya mix units (cm aur m) ko bina convert kiye seedha use kar lena.
Board-Style Important Questions
- Ek right-angled triangle ke legs 5 cm aur 12 cm hain — hypotenuse nikaliye.
- Ek rectangle field ki length 40 m aur breadth 9 m hai — iski diagonal length nikaliye.
- Check kijiye ki 7 cm, 24 cm, 25 cm sides waala triangle right-angled hai ya nahi.
- Ek ladder deewar se 5 m door rakhi hai aur uski height 12 m tak pahunchti hai — ladder ki length nikaliye.
- Baudhayana-Pythagoras theorem ka statement likhiye aur ek diagram ke through explain kijiye.
Aksar Poochhe Jaane Wale Sawaal
Baudhayana-Pythagoras Theorem ka matlab kya hai class 8 Ganita Prakash me?
Ye ek geometric relationship hai jo kehta hai ki kisi bhi right-angled triangle me hypotenuse (right angle ke saamne waali side) ka square, doosri do sides ke squares ke sum ke barabar hota hai. NCERT ne ise Baudhayana ke naam ke saath jodkar rakha hai kyunki ye result Baudhayana ke Sulba Sutra me Pythagoras se pehle hi discover ho chuka tha.
Class 8 maths ka new book name 2026-27 session ke liye kya hai?
NCERT ne Class 8 Maths ke liye purani single-book format hata kar do-part edition launch kiya hai jiska naam Ganita Prakash hai — Part 1 (Chapters 1-7) aur Part 2 (Chapters 8-14), total 14 chapters.
Baudhayana-Pythagoras theorem kis type ke triangles par apply hoti hai?
Ye theorem sirf right-angled triangles (jinme ek angle exactly 90° ho) par apply hoti hai. Non-right-angled triangles ke liye ye relationship valid nahi hai.
Hypotenuse kaise identify karein kisi triangle me?
Hypotenuse hamesha right angle (90°) ke saamne waali side hoti hai, aur triangle ki teeno sides me sabse lambi hoti hai. Baaki do sides ko legs ya base/height kaha jaata hai.
Pythagorean triplets kya hote hain aur kaunse common hain?
Pythagorean triplets teen whole numbers ka set hote hain jo theorem ko satisfy karte hain, jaise 3-4-5, 5-12-13, 8-15-17, 7-24-25. Inhe yaad rakhne se exam me calculation fast ho jaati hai kyunki inke multiples (jaise 6-8-10) bhi apply hote hain.
Ganita Prakash Part 1 aur Part 2 me kitne chapters hain aur is chapter ka number kya hai?
Ganita Prakash total 14 chapters me hai — Part 1 me Chapters 1 se 7 (jaise 'A Square and A Cube', 'We Distribute Yet Things Multiply', 'Tales by Dots and Lines', 'Algebra Play') aur Part 2 me Chapters 8 se 14. 'The Baudhayana-Pythagoras Theorem' Part 2 ka Chapter 9 hai.
Class 8 Maths — Saare Chapters
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