Class 8 Maths Part 2 Chapter 2: The Baudhayana-Pythagoras Theorem (Hinglish Notes)

Ye chapter kis baare mein hai?

Lagbhag 800 BCE mein Baudhayana ne ek Shulbasutra likha, yagya vedi ko sahi aakaar aur maap mein banaane ki kitaab. Banaane walon ko ek square ko doosre se doguna banaana hota tha, ya do squares ke barabar ek square. Unke sawaalon ke peechhe chalte hue chapter geometry ka sabse zaroori nateeja dhoondhta hai: wo theorem jo har right-angled triangle ki teeno bhujaon ko jodta hai.

Square ko doguna aur aadha karna

Square ki bhuja doguni karne se area doguna nahi, chaar guna hota hai. Baudhayana ka jawab hai diagonal par square banaana. Uttar-dakshin aur poorab-pashchim lines purane square ko 2 ek jaise right triangles mein aur naye ko 4 mein kaatti hain, isliye naye ka area doguna hai. Is idea ko ulta chalaao to square aadha hota hai: square kaagaz ke kone beech mein modo, aur beech ke binduon se guzarti silvatein aadhe area ka square banaati hain.

Sankhya √2

Jis right triangle ki do chhoti bhujaayein 1 unit hon, wo unit square ka aadha hai, aur uske hypotenuse par bana square wahi doguna square hai, jiska area 2 hai. To hypotenuse c ke liye c² = 2, aur c = √2. Aam taur par barabar bhujaayein a hon to c² = 2a², yaani c = a√2. Decimals ka varg nikaalkar √2 ko aur kas kar pakadte hain: 1.4 < √2 < 1.5, phir 1.41 < √2 < 1.42, phir 1.414 < √2 < 1.415. Iska decimal kabhi khatm nahi hota, aur jaise Euclid ne lagbhag 300 BCE mein saabit kiya, ise fraction nahi likh sakte. Padho root 2 kya hai?

Do squares jodna: theorem

Agar do alag squares ko ek mein jodna ho? Baudhayana ka niyam: aisa right triangle banao jiski chhoti bhujaayein dono squares ki bhujaayein hon; uske hypotenuse par bane square ka area theek dono ke jod jitna hota hai. Dono squares ko saath rakhkar aur do triangles khiskaakar ye saabit hota hai, jisse Baudhayana ka theorem milta hai: kisi bhi right triangle mein, chhoti bhujaayein a aur b aur hypotenuse c ho to a² + b² = c². Isliye 3 cm aur 4 cm bhujaon wale right triangle ka hypotenuse theek 5 cm hai, kyunki 9 + 16 = 25. Ise Pythagorean theorem bhi kehte hain, Pythagoras ke naam par, jinhone Baudhayana ke lagbhag teen sadi baad ise padha. Padho Baudhayana-Pythagoras theorem kya hai?

Baudhayana triples

Ginti ki sankhyaayein (a, b, c) jinmein a² + b² = c² ho, Baudhayana triples ya Pythagorean triples kehlaati hain. Baudhayana ne khud chhah bataaye, jinmein (3, 4, 5), (5, 12, 13), (7, 24, 25) aur (8, 15, 17) hain. Kisi triple ko kisi bhi ginti ki sankhya se guna karne par naya triple milta hai, jaise (6, 8, 10), isliye ye anginat hain. Jin triples ka koi common factor na ho, jaise (3, 4, 5), ve primitive hain. Ek chaalaak tareeka aur triples deta hai: pehli n odd sankhyaon ka jod n² hota hai, isliye (n − 1)² + (2n − 1) = n², aur jab bhi 2n − 1 khud varg ho, naya triple mil jaata hai. Padho Pythagorean triples kya hain?

300 saal se zyada ka sawaal

Kyunki kai varg do vargon ke jod hote hain, 17th century mein French ganitagya Fermat ne poochha ki kya koi cube do cubes ka jod ho sakta hai, ya koi fourth power do fourth powers ka. Unhone hashiye mein likha ki n ke 2 se bada hone par xⁿ + yⁿ = zⁿ ka koi hal nahi, aur unke paas saboot hai jo hashiye mein nahi samaata. 300 saal se zyada tak koi use nahi dhoondh paaya. 1963 mein 10 saal ke Andrew Wiles ne is sawaal ke baare mein padha aur ise hal karne ki thaani; unhone 1994 mein Fermat’s Last Theorem saabit kiya.

Theorem ka istemaal

Theorem un lambaaiyon ko nikaalta hai jo seedhe naapi nahi ja sakti. Bhaskaracharya ki Lilavati mein ek kamal jheel ke paani se 1 unit upar hai aur hawa se jhukkar 3 unit door paani chhoota hai. Gehraai x ho to dandi x + 1 hai, aur 3² + x² = (x + 1)² se x = 4 milta hai. Isi idea se square ka diagonal, diagonals se rhombus ki bhuja, aur equilateral triangle ki oonchai milti hai.

Saar mein

Har right triangle mein hypotenuse par bana square baaki do bhujaon par bane squares ke jod ke barabar hota hai. Gaayab bhuja nikaalne, ye jaanchne ki teen lambaaiyaan right triangle banaati hain ya nahi, aur triples banaane ke liye a² + b² = c² use karo. Chapter galat label wale teen dabbon ki paheli par khatm hota hai. Har sawaal ke jawab Part 2 Chapter 2 ke NCERT solutions mein hain.