✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 8² + 15² kitna hai?
- 12 aur 18 ka HCF (sabse bada common factor)?
- Kya 17² = 289 hai?
Answers dekho
- 64 + 225 = 289
- 6
- Haan, 17 × 17 = 289
Teeno sahi? Shaabaash! HCF yaad nahi? Koi baat nahi, is post mein “common factor” ka bas itna matlab hai: aisa number jo teeno ko poora-poora baant de.
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🎬 Kahani: Mistri Chacha Ki Rassi
Naye ghar ki neev ban rahi thi. Mistri Chacha ne ek rassi pe 3, 4, 5 haath ki doori pe gaanthein baandhi aur triangle bana ke kona banaya: bilkul 90°!
Golu ne poochha, “Chacha, 3, 4, 5 hi kyun? Koi aur number nahi chalte?”
Chacha hanse, “Bade kamre ke liye main 6, 8, 10 leta hoon. Aur kabhi-kabhi 5, 12, 13 bhi!”
Golu soch mein pad gaya: aise kitne number-jode hain jahan teeno poore number hon aur a² + b² = c² ho? Baudhayana ne iski list 2800 saal pehle di thi! 📜
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📖 Concept: Teeno Sides Poore Number
Step 1: Triple kya hai?
Teen positive poore numbers (a, b, c) jinme a² + b² = c² ho, unhe Baudhayana triple kehte hain. Inhe Baudhayana–Pythagoras triple, Pythagorean triple ya right-angled triangle triple bhi kehte hain.
WHY khaas? Zyaadatar right triangles mein koi na koi side √ wali (adhoori) hoti hai. Triples mein teeno poori, isliye rassi se naapna aasaan.
Step 2: Baudhayana ki list (Śulba-Sūtra 1.13)
| Triple | Check |
|---|---|
| (3, 4, 5) | 9 + 16 = 25 |
| (5, 12, 13) | 25 + 144 = 169 |
| (8, 15, 17) | 64 + 225 = 289 |
| (7, 24, 25) | 49 + 576 = 625 |
| (12, 35, 37) | 144 + 1225 = 1369 |
| (15, 36, 39) | 225 + 1296 = 1521 |
Step 3: Scaling: ek triple se anant triples
(3, 4, 5) ko 2 se guna: (6, 8, 10) → 36 + 64 = 100 ✅. 3 se: (9, 12, 15) ✅. 10 se: (30, 40, 50) ✅. 100 se: (300, 400, 500) ✅.
General: (a, b, c) triple hai to (ka, kb, kc) bhi, kyunki (ka)² + (kb)² = k²(a² + b²) = k²c² = (kc)².
WHY aisa? Triangle ko “zoom” karne jaisa hai: shape same, bas bada. Kone wahi rehte hain, to 90° bhi wahi.
Step 4: Primitive triple
Jis triple ke teeno numbers ka 1 ke alaawa koi common factor na ho, use primitive kehte hain. (3, 4, 5) primitive hai; (9, 12, 15) nahi (3 common).
Ulta bhi: (9, 12, 15) ko 3 se bhaag do → (3, 4, 5), phir se triple! Har non-primitive triple kisi primitive ka scaled roop hai.
WHY yeh farak? Primitive triples hi “asli naye” hain. Baaki sab unke zoom kiye hue roop.
Step 5: 20 tak ke saare triples
(3, 4, 5), (6, 8, 10), (5, 12, 13), (9, 12, 15), (8, 15, 17), (12, 16, 20). Inme primitive: (3, 4, 5), (5, 12, 13), (8, 15, 17).
WHY list banana? Pattern dikhne lagte hain: har primitive triple mein c odd hai aur ek leg even. Agle post mein triples banane ki trick!
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🔐 Deep Concept: Triple Ki Chhupi Pehchaan
Primitive triples mein kuch niyam hamesha dikhte hain:
- Ek leg even, ek odd, aur hypotenuse odd.
- Kisi na kisi leg mein 3 ka factor, kisi leg mein 4 ka factor.
- Teeno mein se koi na koi 5 se divide hota hai.
Check (8, 15, 17): 8 even (4 ka multiple), 15 odd (3 aur 5 ka multiple), 17 odd ✅. Check (20, 21, 29): 20 mein 4 aur 5, 21 mein 3 ✅.
Isliye kisi bhi triple ke do legs ka guna hamesha 12 se divide hota hai! (3 × 4 = 12, 5 × 12 = 60, 8 × 15 = 120) 🔐
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✏️ Solved Examples
Example 1 (Easy): Kya (20, 21, 29) triple hai?
400 + 441 = 841 = 29² ✅ Haan. Common factor? 20, 21 ka koi common factor nahi, to primitive bhi.
WHY sahi hai? 29 × 29 = 841 ✅
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Example 2 (Medium): (5, 12, 13) ke 4 scaled triples likho aur batao primitive hain ya nahi.
(10, 24, 26), (15, 36, 39), (20, 48, 52), (25, 60, 65). Teeno mein k common, to koi primitive nahi.
WHY sahi hai? (15, 36, 39) Baudhayana ki list mein hai, aur 3 × (5, 12, 13) hi hai ✅
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Example 3 (NCERT style): (9, 12, 15) ke teeno numbers 3 se divide hote hain. Kya (9/3, 12/3, 15/3) triple hai? Kyun?
(3, 4, 5) ✅. General: agar a² + b² = c² aur f teeno ko baant de, to (a/f)² + (b/f)² = (a² + b²)/f² = c²/f² = (c/f)².
WHY sahi hai? Bhaag dena = ulta zoom. Shape wahi, kona wahi 90° ✅
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Chacha ki seekh: “3, 4, 5 yaad rakho, baaki uske bete-pote (6, 8, 10…) apne aap aa jaayenge!” 🎉
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🏠 Real Life Mein
- 🧱 Right angle banana: 3-4-5 rassi, ya bade kaam ke liye 6-8-10, 9-12-15.
- 🏏 Khel maidaan: Court ke kone sahi 90° hain ya nahi, 3-4-5 ki naap se check.
- 📐 Carpenter: Almari ka frame square hai? Kone se 30 cm aur 40 cm pe nishaan, beech 50 cm ho to sahi.
- 🖥️ Screen: Kai purani screens 4 : 3 ki thi, unka diagonal 5 ke anupaat mein!
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🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Teeno poore number + a² + b² = c² | ✅ Triple | Definition |
| Koi number fraction/decimal | ❌ Triple nahi | Poore number chahiye |
| Common factor hai | ✅ Triple, par primitive nahi | Kisi primitive ka zoom |
| Bada triple chahiye | ✅ Kisi triple ko k se guna karo | (ka, kb, kc) bhi triple |
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❌→✅ Worked Mistake
Sawaal: (3, 4, 5) triple hai. Kya (4, 5, 6) bhi triple hai? (Har number 1 badhaaya)
❌ Galat: “Haan, pattern wahi hai.”
Galti kya hai? 16 + 25 = 41, par 6² = 36. 41 ≠ 36 ❌. Jodne se triple nahi banta, sirf guna karne se.
✅ Sahi:
- (3 + 1, 4 + 1, 5 + 1) → triple nahi.
- (3 × 2, 4 × 2, 5 × 2) = (6, 8, 10) → triple ✅
Yaad rakho: Triple ko scale karo (guna), shift mat karo (jodo).
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📝 10 MCQs
Q1. Kaun sa triple hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 41 ≠ 36.
- (b) sahi: 36 + 64 = 100 ✅
- (c) galat: 13 ≠ 16.
- (d) galat: 61 ≠ 49.
Q2. Kaun sa primitive triple hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: 2 common.
- (b) galat: 3 common.
- (c) sahi: Koi common factor nahi ✅
- (d) galat: 10 common.
Q3. (3, 4, 5) × 7 = ?
Solution dekho
Sahi jawab: (b)
- (a) galat: 7 joda, guna nahi kiya.
- (b) sahi: Sahi guna ✅
- (c) galat: 36 galat, 5 × 7 = 35.
- (d) galat: Sirf ek number guna kiya.
Q4. 20 tak kitne primitive triples hain (teeno ≤ 20)?
Solution dekho
Sahi jawab: (b)
- (a) galat: Ek chhoot gaya.
- (b) sahi: (3,4,5), (5,12,13), (8,15,17) ✅
- (c) galat: 5 nahi.
- (d) galat: 6 kul triples hain, primitive nahi.
Q5. (12, 35, ?) triple ka teesra number?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 144 + 1225 = 1369 = 37² ✅
- (b) galat: 47² = 2209.
- (c) galat: 43² = 1849.
- (d) galat: 41² = 1681.
Q6. Agar (a, b, c) triple hai to (ka, kb, kc) kyun?
Solution dekho
Sahi jawab: (a)
- (a) sahi: k²(a² + b²) = k²c² ✅
- (b) galat: Jodne se nahi chalta.
- (c) galat: Hamesha chalta hai.
- (d) galat: Har positive k pe.
Q7. (15, 36, 39) kiska scaled roop hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 5 × 3 = 15 par 4 × 3 = 12 ≠ 36.
- (b) sahi: ÷ 3 → (5, 12, 13) ✅
- (c) galat: Factor nahi milta.
- (d) galat: Factor nahi milta.
Q8. Primitive triple mein hypotenuse…
Solution dekho
Sahi jawab: (b)
- (a) galat: Primitive mein c even nahi hota.
- (b) sahi: 5, 13, 17, 25, 29… ✅
- (c) galat: Niyam hai: odd.
- (d) galat: 5 sirf (3, 4, 5) mein.
Q9. Kya (0.3, 0.4, 0.5) Baudhayana triple hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: Right triangle to banta hai, par triple poore numbers ka hota hai.
- (b) sahi: Poore number zaroori ✅
- (c) galat: Primitive ki baat hi nahi.
- (d) galat: Definition se saaf.
Q10. (7, 24, 25) ke legs ka guna 12 se divide hota hai?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 7 × 24 = 168 = 12 × 14 ✅
- (b) galat: 168 ÷ 12 = 14, poora.
- (c) galat: 12 se bhi hota hai.
- (d) galat: Guna ki baat hai.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Triple aur right triangle mein kya farak?
Har triple ek right triangle deta hai, par har right triangle triple nahi (jaise 1, 1, √2).
2. Kitne triples hain?
Anant! Sirf (3, 4, 5) ko 2, 3, 4… se guna karte jao.
3. Primitive triples bhi anant hain?
Haan! Agle post mein ek trick se naye-naye primitive triples banayenge.
4. (5, 12, 13) aur (12, 5, 13) alag hain?
Same triangle hai, bas legs ki jagah badli. Aam taur pe chhota pehle likhte hain.
5. Common factor kaise dhoondhein?
Teeno ka HCF nikaalo. HCF = 1 to primitive.
6. Kya c hamesha sabse bada hota hai?
Haan, c² = a² + b², to c, a aur b dono se bada.
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy Kya (9, 40, 41) triple hai?
Solution dekho
81 + 1600 = 1681 = 41² ✅ Haan.
Q2. Easy (8, 15, 17) × 2?
Solution dekho
(16, 30, 34).
Q3. Easy Kya (10, 24, 26) primitive hai?
Solution dekho
Nahi, 2 common.
Q4. Easy (6, 8, 10) ka primitive roop?
Solution dekho
÷ 2 → (3, 4, 5).
Q5. Medium (7, 24, 25) ke 3 scaled triples?
Solution dekho
(14, 48, 50), (21, 72, 75), (28, 96, 100).
Q6. Medium Kya (11, 60, 61) triple hai, aur primitive?
Solution dekho
121 + 3600 = 3721 = 61² ✅, primitive.
Q7. Medium 30 tak ke saare triples jinme c = 25 ho?
Solution dekho
(7, 24, 25) aur (15, 20, 25).
Q8. NCERT style 20 tak ke saare triples likho.
Solution dekho
(3,4,5), (6,8,10), (5,12,13), (9,12,15), (8,15,17), (12,16,20).
Q9. NCERT style Kya (30, 40, 50) aur (300, 400, 500) triples hain?
Solution dekho
Haan, (3, 4, 5) × 10 aur × 100.
Q10. Medium Kya (16, 63, 65) triple hai?
Solution dekho
256 + 3969 = 4225 = 65² ✅
Q11. Medium (20, 21, 29) ka doosra scaled triple jisme c = 87 ho?
Solution dekho
× 3 → (60, 63, 87).
Q12. NCERT style f teeno ko baant de to (a/f, b/f, c/f) triple kyun?
Solution dekho
(a² + b²)/f² = c²/f² ✅
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🧠 HOTS (Sochne Wale Sawaal)
Q1. Kya koi triple hai jisme a = b ho?
Solution dekho
a² + a² = 2a² = c² → c = a√2, jo poora nahi. Nahi.
Q2. Kya (3, 4, 5) se jodne-ghataane se doosra triple ban sakta hai?
Solution dekho
Shift se aam taur pe nahi (4, 5, 6 fail). Guna hi pakka tareeka hai.
Q3. Golu: “Har triple mein ek number 5 ka multiple hai.” Check karo (8, 15, 17), (20, 21, 29), (12, 35, 37).
Solution dekho
15 ✅, 20 ✅, 35 ✅. Sach hai!
Q4. Kisi triple mein c = 13 ho to a, b?
Solution dekho
Sirf (5, 12, 13) (169 − 144 = 25).
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🏆 Olympiad Corner
Q1. Triple ke legs ka guna hamesha kis number se divide hota hai?
Solution dekho
12 (ek leg mein 3, ek mein 4 ka factor; scaled triples mein bhi).
Q2. Area aur perimeter barabar wale right triangles (poori sides) dhoondho.
Solution dekho
(6, 8, 10): area 24, perimeter 24 ✅. (5, 12, 13): area 30, perimeter 30 ✅.
Q3. Kya koi triple hai jisme c = 2 ho?
Solution dekho
Nahi. 1² + 1² = 2 ≠ 4, aur legs ≥ 1. Sabse chhota c = 5.
Q4. (33, 56, ?) triple pura karo.
Solution dekho
1089 + 3136 = 4225 = 65². 65.
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💗 CharuMam Motivation
Mistri Chacha ne kabhi “theorem” shabd nahi suna, phir bhi roz usse ghar banaate hain. Ganit kitaabon mein hi nahi, haathon ke hunar mein bhi basa hai. 🧱
Jo aap padh rahe ho, woh kisi na kisi ki roti-rozi aur sapnon ka ghar banata hai. Isliye mann se padho! 🏡
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🟢 Master Recap
- Baudhayana triple: positive poore numbers (a, b, c) jinme a² + b² = c².
- Baudhayana ki list: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (12,35,37), (15,36,39).
- (a, b, c) triple → (ka, kb, kc) bhi triple.
- Primitive: 1 ke alaawa koi common factor nahi.
- Common factor f ho to (a/f, b/f, c/f) bhi triple.
- 20 tak: 6 triples, jinme 3 primitive.
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👉 Aage Kya?
Naye primitive triples kaise banayein? Agle post mein odd numbers ka jaadu se triples banayenge, aur sunenge ek 350 saal purani paheli ki kahani: Fermat ka Last Theorem! 🧩
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬