Baudhayana-Pythagoras theorem kya hai? (a² + b² = c²)
Short mein jawab
Baudhayana-Pythagoras theorem kehta hai ki kisi right-angled triangle mein, chhoti bhujaayein a aur b aur hypotenuse c ho, to a² + b² = c²: hypotenuse par bana square baaki do squares ke milaakar jitna bada hota hai. 3 cm aur 4 cm bhujaon ke liye c² = 9 + 16 = 25, yaani hypotenuse theek 5 cm hai.
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Do alag squares jodna
Square ko doguna karna aasaan hai: uske diagonal par square banao. Par agar a aur b bhujaon ke do alag squares ko ek aise square mein jodna ho jiska area dono ka jod ho? Apne Shulbasutra (verse 1.12) mein Baudhayana ne jawab diya: aisa right-angled triangle banao jiski chhoti bhujaayein a aur b hon. Uske hypotenuse par bana square theek dono squares ke milaakar area jitna hota hai. Jab a aur b barabar hon, to ye doguna karne wala niyam hi hai, kyunki tab hypotenuse square ka diagonal hota hai.
Theorem
Symbols mein Baudhayana ka niyam hi theorem hai: jis right-angled triangle ki chhoti bhujaayein a aur b hon aur hypotenuse c, usmein a² + b² = c². Baudhayana ne sabse pehle lagbhag 800 BCE mein ise is saamaanya roop mein likha. Greek darshanik-ganitagya Pythagoras ne, jo lagbhag teen sadi baad hue, ise padha, isliye ye Pythagorean theorem ke naam se mashhoor hai. Baudhayana-Pythagoras theorem kehne se saaf ho jaata hai ki kaun sa theorem hai.
Ye sach kyon hai
Baudhayana ka verse 2.1 ek saboot ki taraf ishaara karta hai. b bhuja ka square aur a bhuja ka square saath-saath rakho, neeche ek line mein. Base ke baayen sire se a unit door ek bindu lagao, aur wahan se bade square ke upar-baayen kone tak aur chhote square ke upar-daayen kone tak kaato. Har kaat a aur b bhujaon wale right triangle ka hypotenuse c hai. Ab kate hue dono triangles ko upar ki khaali jagahon mein khiskaao. Teeno tukde milkar c bhuja ka square bana dete hain. Iske chaaron kone right angles hain, kyunki har kone par ek right triangle ke do acute angles, x aur 90° − x, milkar 90° bante hain. Kuch joda ya ghataaya nahi gaya, isliye a² + b² = c².
Gaayab bhuja nikaalna
Theorem do aasaan niyam deta hai. Hypotenuse ke liye vargon ko jodo aur square root lo: c = √(a² + b²). 3 cm aur 4 cm bhujaon wala right triangle banaao to hypotenuse lagbhag 5 cm naapta hai; theorem kehta hai ki ye theek 5 cm hai, kyunki 9 + 16 = 25. Chhoti bhuja ke liye ghataao: ek bhuja 8 cm aur hypotenuse 17 cm ho, to teesri bhuja √(289 − 64) = √225 = 15 cm. Jawab poori sankhya hona zaroori nahi: 5 aur 7 bhujaayein c = √74 deti hain, lagbhag 8.60.
Teen guna aur paanch guna square
Baudhayana ne ye bhi dikhaaya ki diye gaye square se teen guna bada square kaise banaayein (verse 1.10). Aisa right triangle lo jiski chhoti bhujaayein square ki bhuja a aur uska diagonal a√2 hon. Tab c² = a² + 2a² = 3a², isliye c par bana square teen guna area ka hai. Chhoti bhujaayein a aur 2a lo to c² = a² + 4a² = 5a², paanch guna area.
Kahan kaam aata hai
Theorem se 5 cm bhuja wale square ka diagonal 5√2, lagbhag 7.07 cm milta hai; 24 aur 70 diagonals wale rhombus ki bhuja √(12² + 35²) = 37; aur Bhaskaracharya ke kamal wale sawaal mein jheel ki gehraai 4 unit. Poori sankhya wale jawaab Pythagorean triples kya hain? mein aur √2 root 2 kya hai? mein padho. Poora chapter Chapter 2 ke notes mein hai, aur har exercise NCERT solutions mein.
Practice karo
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Right triangle ki chhoti bhujaayein 5 cm aur 12 cm hain. Hypotenuse kitna hai?
Jawab dekho
Jawab: (a) 13 cm
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Right triangle ki ek bhuja 8 cm aur hypotenuse 17 cm hai. Teesri bhuja kitni hai?
Jawab dekho
Jawab: (b) 15 cm
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9 cm aur 12 cm ke rectangle ka diagonal kitna hai?
Jawab dekho
Jawab: (a) 15 cm
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a = 7 aur b = 12 ho to gaayab bhuja nikaalo.
Jawab dekho
Jawab: c² = 7² + 12² = 49 + 144 = 193, isliye c = √193, lagbhag 13.89.
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Kya hypotenuse hamesha right triangle ki sabse lambi bhuja hoti hai?
Jawab dekho
Jawab: Haan. c² = a² + b², jo a² se bhi bada hai aur b² se bhi, kyunki har varg positive hai, isliye c, a aur b dono se lamba hai.
Students ke sawal
Baudhayana-Pythagoras theorem kya kehta hai?
Har right-angled triangle mein hypotenuse par bana square theek utna bada hota hai jitne baaki do bhujaon par bane squares milaakar. Symbols mein a² + b² = c², jahan c hypotenuse hai, yaani right angle ke saamne wali bhuja.
Baudhayana ne theorem kaise bataaya?
Apne Shulbasutra (verse 1.12) mein unhone kaha ki rectangle ke diagonal par bana square uski do bhujaon par bane squares ke milaakar area jitna hota hai. Yahi saamaanya theorem hai, Pythagoras se lagbhag teen sadi pehle likha gaya.
a² + b² = c² kaise saabit karein?
b aur a bhujaon ke squares saath-saath rakho. Base par a unit door ke bindu se do konon tak kaato; har kaat a-b right triangle ka hypotenuse c hai. Kate hue do triangles khiskaane par c bhuja ka square poora bhar jaata hai, isliye a² + b² = c².
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