✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 5² + 12² kitna hai?
- √169 kitna hai?
- Right triangle mein hypotenuse ke alaawa sides ko kya kehte hain?
Answers dekho
- 25 + 144 = 169
- 13
- Legs (90° banane wali sides)
Teeno sahi? Badhiya! Theorem dobara dekhna ho to Post 4 — Baudhayana–Pythagoras Theorem Kya Hai? padh lijiye.
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🎬 Kahani: Patang Ki Dor
Chhat pe Rohan patang uda raha tha. Dor 17 m khuli thi aur patang seedha upar nahi, thoda door thi. Priya ne neeche se dekha: patang Rohan se zameen pe 8 m door wali jagah ke theek upar thi.
“Patang kitni oonchi hai?” Rohan ne poochha. Priya muskuraayi, “Dor, zameen aur oonchaai ek right triangle bana rahe hain. Hypotenuse pata hai (17), ek leg pata hai (8). Doosri leg nikaal lete hain!” 🪁
(Maan lo dor bilkul seedhi tani hai aur Rohan ke haath ki oonchaai chhod do.)
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📖 Concept: Teen Mein Se Do Pata Hon, Teesri Nikaalo
Step 1: Pehle pehchaano, hypotenuse kaun hai
Sabse pehle dekho 90° kahan hai. Uske saamne wali side = c. Baaki do = a, b.
WHY pehle yeh? Formula mein c ki jagah fixed hai. Galat side ko c maana to jawab galat.
Step 2: Hypotenuse chahiye → jodo
c = √(a² + b²)
Jaise legs 5, 12: c = √(25 + 144) = √169 = 13.
Step 3: Leg chahiye → ghatao
a² + b² = c² ko ulta karo:
b = √(c² − a²)
Patang: b = √(17² − 8²) = √(289 − 64) = √225 = 15 m oonchi! 🪁
WHY ghatao? Hypotenuse wala square sabse bada hai aur woh dono chhote squares ka jod hai. Ek chhota square nikaal do, doosra bach jaata hai.
Step 4: Jawab poora number na ho to?
Legs 5 aur 7: c = √(25 + 49) = √74. 8² = 64, 9² = 81, to 8 < √74 < 9. 8.6² = 73.96, 8.7² = 75.69, to c ≈ 8.6.
WHY gherna? Har √ poora nahi aata. Bounds se pata chalta hai jawab kahan hai, aur ek decimal tak anumaan pakka.
Step 5: Hamesha check karo
Jawab milne ke baad: (1) hypotenuse sabse lamba hai? (2) chhoti sides ke square jod ke c² aata hai?
WHY? Agar kisi leg ka jawab hypotenuse se bada aaya, to pakka galti hai (shayad jodna tha, ghataaya, ya ulta).
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🔐 Deep Concept: Square Ka 3 Guna, 5 Guna Area
Post 1 mein double area wala square diagonal se bana. Teen guna kaise? Śulba-Sūtra (verse 1.10) ka idea: theorem se squares jodo!
- 3 guna: Side s wale square ka diagonal = s√2 (uska square 2s²). Ab ek right triangle banao jiske legs s aur diagonal hon. Hypotenuse² = s² + 2s² = 3s². Is hypotenuse pe bana square = teen guna! ✅
- 5 guna: Legs s aur 2s lo. Hypotenuse² = s² + 4s² = 5s² ✅
- 4 guna to aasaan: side 2s.
Aise hi aage badhte raho: har naya hypotenuse aur s se agla right triangle, aur area 2, 3, 4, 5, 6… guna! Yeh ek “ghoomti seedhi” (spiral) banata hai. 🌀🔐
sirf charumam.com readers ke liye 🔐
✏️ Solved Examples
Example 1 (Easy): Legs 5 cm aur 12 cm. Hypotenuse?
c² = 25 + 144 = 169, c = 13 cm.
WHY sahi hai? 13 sabse lamba aur 169 = 25 + 144 ✅
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Example 2 (Medium): Chhoti side 8 cm aur hypotenuse 17 cm. Teesri side?
b² = 17² − 8² = 289 − 64 = 225. b = 15 cm.
WHY sahi hai? 8² + 15² = 64 + 225 = 289 = 17² ✅
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Example 3 (NCERT style): Right triangle mein a = 9, c = 15 (c hypotenuse). b?
b² = 225 − 81 = 144, b = 12.
WHY sahi hai? 9, 12, 15 = 3, 4, 5 ka 3 guna ✅
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Example 4 (NCERT style): Legs 7 aur 12. Hypotenuse ko ek decimal tak batao.
c² = 49 + 144 = 193. 13² = 169, 14² = 196. 13.8² = 190.44, 13.9² = 193.21.
To 13.8 < c < 13.9 (c ≈ 13.89).
WHY sahi hai? 193 aur 196 bahut paas hain, to c lagbhag 14 se thoda kam ✅
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Priya ki trick: “Hypotenuse chahiye? Jodo. Leg chahiye? Ghatao. Phir √ lo!” 🎉
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🏠 Real Life Mein
- 🪁 Patang/dhwaj: Dor ki lambaai aur zameen pe doori se oonchaai.
- 🪜 Seedhi: Seedhi ki lambaai aur deewar se doori se, seedhi deewar pe kitni oonchi pahunchegi.
- 🏗️ Ramp: Wheelchair ramp ki oonchaai aur lambaai se uski tircha lambaai.
- 📶 Mobile tower ki taar: Tower ki oonchaai aur zameen pe khoonti ki doori se taar ki lambaai.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Do legs pata, hypotenuse chahiye | ✅ c = √(a² + b²) | Squares jodo |
| Hypotenuse aur ek leg pata | ✅ b = √(c² − a²) | Squares ghatao |
| Leg ka jawab hypotenuse se bada aaya | ❌ Galti hai | Hypotenuse hamesha sabse lamba |
| Triangle right-angled nahi | ❌ Formula nahi | Theorem sirf 90° pe |
— 📌 charumam.com
❌→✅ Worked Mistake
Sawaal: Hypotenuse 10 cm, ek leg 6 cm. Doosri leg?
❌ Galat: “b² = 10² + 6² = 136, b ≈ 11.66 cm.”
Galti kya hai? Hypotenuse pata tha, to ghatana tha. Aur jawab 11.66 hypotenuse (10) se bada aa gaya, jo ho hi nahi sakta ❌
✅ Sahi:
- b² = 100 − 36 = 64.
- b = 8 cm. Check: 36 + 64 = 100 ✅
Yaad rakho: Pehle c pehchaano, phir tay karo jodna hai ya ghatana.
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📝 10 MCQs
Q1. Legs 9 aur 40. Hypotenuse?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 81 + 1600 = 1681 = 41² ✅
- (b) galat: 9 + 40, galat.
- (c) galat: 40 − 9, galat.
- (d) galat: 9 × 40, galat.
Q2. Hypotenuse 13, ek leg 5. Doosri leg?
Solution dekho
Sahi jawab: (b)
- (a) galat: 13 − 5, sides ghataayi, squares nahi.
- (b) sahi: 169 − 25 = 144 = 12² ✅
- (c) galat: 13 + 5, galat.
- (d) galat: Yeh jodne se aata, par c pata tha.
Q3. Leg chahiye ho to formula?
Solution dekho
Sahi jawab: (b)
- (a) galat: Jodne se hypotenuse se bada aayega.
- (b) sahi: Bada square − chhota square ✅
- (c) galat: Sides seedhe nahi ghatti.
- (d) galat: Yeh to alag hi cheez hai.
Q4. Legs 5 aur 7 ka hypotenuse kin poore numbers ke beech?
Solution dekho
Sahi jawab: (b)
- (a) galat: 7² = 49, 74 se kam, par 8² = 64 bhi kam.
- (b) sahi: 64 < 74 < 81 ✅
- (c) galat: 9² = 81 pehle hi zyada.
- (d) galat: Bahut bada.
Q5. Patang: dor 17 m, zameen pe doori 8 m. Oonchaai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 17 − 8, galat tareeka.
- (b) sahi: √(289 − 64) = √225 = 15 ✅
- (c) galat: 17 + 8, galat.
- (d) galat: Jodna nahi tha.
Q6. Legs 1.5 aur 3.5. c² kitna?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 2.25 + 12.25 = 14.5 ✅
- (b) galat: (1.5 + 3.5)² = 25, galat.
- (c) galat: 1.5 + 3.5, galat.
- (d) galat: Sirf 3.5² hai.
Q7. Square ka 3 guna area chahiye. Legs kya lein?
Solution dekho
Sahi jawab: (b)
- (a) galat: s² + s² = 2s², double.
- (b) sahi: s² + 2s² = 3s² ✅
- (c) galat: s² + 4s² = 5s².
- (d) galat: Yeh 8 guna dega.
Q8. Legs s aur 2s se hypotenuse pe bana square kitne guna?
Solution dekho
Sahi jawab: (c)
- (a) galat: 3 ke liye s aur s√2.
- (b) galat: 4 ke liye side 2s.
- (c) sahi: s² + 4s² = 5s² ✅
- (d) galat: 6 ke liye aur ek kadam.
Q9. Legs 8 aur 12. c ≈ ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: √208 ≈ 14.42 ✅
- (b) galat: 8 + 12 galat.
- (c) galat: 10² = 100, kam.
- (d) galat: 8 × 12 galat.
Q10. Kaun sa jawab pakka galat hai? (hypotenuse 10)
Solution dekho
Sahi jawab: (c)
- (a) galat: 6 < 10, sambhav.
- (b) galat: 8 < 10, sambhav.
- (c) sahi: Leg hypotenuse se bada nahi ho sakta ✅
- (d) galat: 9 < 10, sambhav (doosri leg √19).
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Kaise yaad rakhein ki jodna hai ya ghatana?
Hypotenuse chahiye → jodo. Hypotenuse diya hai → ghatao.
2. √ poora na aaye to kya likhein?
Exact roop (jaise √74) ya bounds/approx (≈ 8.6). Sawaal jo maange.
3. Kya theorem seedhi wale sawaal mein chalega?
Haan, agar deewar zameen se 90° pe hai, jo aam taur pe maana jaata hai.
4. Legs ki jagah badal sakte hain?
Haan, a aur b koi bhi ho sakti hain. Bas c hypotenuse rahe.
5. Units kaise likhein?
Sab sides same unit mein ho (cm ya m). Jawab bhi usi unit mein.
6. Square ka 3 guna area kyun chahiye hota tha?
Vedi ke alag-alag yagyon ke liye alag naap chahiye the. Śulba-Sūtra mein aise kai constructions hain.
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy Legs 15 aur 20. Hypotenuse?
Solution dekho
225 + 400 = 625, 25.
Q2. Easy Hypotenuse 25, leg 7. Doosri?
Solution dekho
625 − 49 = 576, 24.
Q3. Easy Legs 12 aur 16?
Solution dekho
144 + 256 = 400, 20.
Q4. Easy Hypotenuse 5, leg 3?
Solution dekho
4.
Q5. NCERT style Legs 8 aur 12, c ko ek decimal tak ghero.
Solution dekho
c² = 208. 14.4² = 207.36, 14.5² = 210.25. 14.4 < c < 14.5.
Q6. NCERT style Legs 1.5 aur 3.5, c ko ghero.
Solution dekho
c² = 14.5. 3.8² = 14.44, 3.9² = 15.21. 3.8 < c < 3.9.
Q7. NCERT style 5 cm side ke square ka 3 guna area wala square kaise banaoge?
Solution dekho
Legs 5 cm aur 5√2 (diagonal) ka right triangle; hypotenuse² = 25 + 50 = 75. Us pe square = 75 cm² ✅
Q8. Medium 10 m seedhi, deewar se 6 m door. Deewar pe kitni oonchi?
Solution dekho
√(100 − 36) = 8 m.
Q9. Medium Tower 12 m ooncha, khoonti 5 m door. Taar?
Solution dekho
√(144 + 25) = 13 m.
Q10. Medium Rectangle 6 × 8 ka diagonal?
Solution dekho
10.
Q11. Medium Legs 2 aur 3. c²? c kin poore numbers ke beech?
Solution dekho
13; 3 < c < 4.
Q12. NCERT style Hypotenuse 2.5, leg 1.5. Doosri?
Solution dekho
6.25 − 2.25 = 4, 2.
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🧠 HOTS (Sochne Wale Sawaal)
Q1. Ek right triangle mein legs barabar hain aur hypotenuse 8 cm. Area?
Solution dekho
2a² = 64 → a² = 32. Area = ½a² = 16 cm².
Q2. Legs a aur b. Agar sirf a ko double karein to kya c bhi double hoga?
Solution dekho
Nahi. c² = 4a² + b², jo 4c² nahi (jab tak b = 0 na ho).
Q3. Dor 17 m hi rahe, par Priya 15 m door chali jaaye (patang uske upar). Oonchaai?
Solution dekho
√(289 − 225) = √64 = 8 m. Doori aur oonchaai ki jagah badal gayi!
Q4. Ek leg 7, hypotenuse aur doosri leg mein 1 ka farak. Sides?
Solution dekho
c² − b² = 49, (c − b)(c + b) = 49, c − b = 1 → c + b = 49 → c = 25, b = 24.
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🏆 Olympiad Corner
Q1. Square ka 6 guna area chahiye. Kaun se legs?
Solution dekho
s aur s√5 (5 guna wale ka side) → s² + 5s² = 6s². Ya s√2 aur 2s → 2s² + 4s² = 6s².
Q2. Right triangle: legs mein farak 7, hypotenuse 13. Legs?
Solution dekho
a² + (a + 7)² = 169 → try a = 5: 25 + 144 = 169 ✅. Legs 5 aur 12.
Q3. Cuboid 3 × 4 × 12 ka sabse lamba diagonal (andar se)?
Solution dekho
Pehle 3, 4 → 5. Phir 5, 12 → 13.
Q4. Seedhi 13 m, neeche 5 m door. Neeche ka sira 7 m khiska to oopar kitna khiskega?
Solution dekho
Pehle oonchaai 12. Ab neeche 12 m door → oonchaai √(169 − 144) = 5. Upar ka sira 7 m neeche aaya.
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💗 CharuMam Motivation
Priya ne patang ki oonchaai naapne ke liye chhat se kood ke naapa nahi, dimaag se naapa! 🪁
Ganit hume woh dikhata hai jo aankh se nahi dikhta. Jab koi cheez “pahunch se bahar” lage, ruk ke socho: kaun si do cheezein pata hain? Teesri apne aap mil jaayegi! 💡
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🟢 Master Recap
- Pehle hypotenuse pehchaano (90° ke saamne).
- Hypotenuse chahiye: c = √(a² + b²) (jodo).
- Leg chahiye: b = √(c² − a²) (ghatao).
- √ poora na ho to bounds se ghero (jaise 8.6 < √74 < 8.7).
- Check: hypotenuse sabse lamba, aur squares ka jod mile.
- 3 guna square: legs s aur s√2; 5 guna: s aur 2s.
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👉 Aage Kya?
Kuch triangles ki teeno sides poore numbers hoti hain, jaise 3, 4, 5. Inhe kehte hain Baudhayana triples! Agle post mein inki duniya, scaling aur primitive triples. 🔢
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬