✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- Legs 12 aur 35 ka hypotenuse?
- Rhombus ke diagonals ek-doosre ko kaise kaatte hain?
- Equilateral triangle ki sides kaisi hoti hain?
Answers dekho
- 144 + 1225 = 1369 = 37²
- Beech se aadha-aadha, aur 90° pe
- Teeno barabar, har kona 60°
Teeno sahi? Ek dum ready ho! Theorem aur missing side yaad nahi? Post 5 dekh lijiye.
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🎬 Kahani: Bhaskaracharya Ka Kamal
Lagbhag 900 saal pehle Bhaskaracharya (Bhaskara II) ne ganit ki ek kitaab likhi (kehte hain apni beti Lilavati ke naam pe): Līlāvatī. Usme ek sundar sawaal hai:
Ek jheel mein kamal (lotus) khila hai. Uska phool paani se 1 haath upar hai. Hawa chali, aur kamal jhuk ke paani ko chhoone laga, apni purani jagah se 3 haath door. Jheel kitni gehri hai? 🪷
Golu bola, “Isme triangle kahan hai?” Priya muskuraayi, “Dhyaan se dekho: dandi (stem), gehraai aur 3 haath ki doori!”
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📖 Concept: Har Jagah Chhupe Right Triangles
Step 1: Kamal ka sawaal
Gehraai = x maano. Dandi = x + 1. Jhukne ke baad dandi hypotenuse ban jaati hai, gehraai aur 3 haath legs.
3² + x² = (x + 1)² → 9 + x² = x² + 2x + 1 → 9 = 2x + 1 → x = 4. Jheel 4 haath gehri, dandi 5 haath. (3, 4, 5) phir se! 🎉
WHY x² kat gaya? Dono taraf x² hai, to wo barabar hokar hat jaata hai, aur seedha sa equation bachta hai.
Step 2: Square aur rectangle ka diagonal
Diagonal rectangle ko do right triangles mein baantta hai. Diagonal = √(l² + b²). 5 cm square: √50 = 5√2 ≈ 7.07 cm.
Integer diagonal wale rectangles: 3 × 4 (5), 6 × 8 (10), 5 × 12 (13), 8 × 15 (17), 7 × 24 (25). Bas triples se!
Step 3: Rhombus ki side
Rhombus ke diagonals ek-doosre ko aadha-aadha aur 90° pe kaatte hain, to 4 right triangles bante hain. Diagonals 24 aur 70: legs 12 aur 35, side = √(144 + 1225) = √1369 = 37.
Step 4: Equilateral triangle ka area
Side 6. Upar ke kone se neeche seedhi line (altitude) daalo, woh neeche ki side ko aadha karti hai: 3 aur 3. Height = √(36 − 9) = √27 ≈ 5.2. Area = ½ × 6 × √27 ≈ 15.6 sq. units (exact: 9√3).
WHY altitude? Area ke liye height chahiye, aur altitude equilateral triangle ko do right triangles mein baant deti hai.
Step 5: Do squares ka farak
7 aur 5 side ke squares ka farak = 49 − 25 = 24. Aisa square banane ke liye: hypotenuse 7, ek leg 5 ka right triangle banao; doosri leg² = 24. Us leg pe square = 24 ✅
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🔐 Deep Concept: Dot Grid Pe Kaun Se Squares Ban Sakte Hain?
Dot paper pe sirf dots ko kone banake square banao. Seedhe squares: area 1, 4, 9… Tirchhe squares: side ek “a aage, b upar” wali line, to area = a² + b².
- Area 2: 1² + 1² ✅
- Area 5: 1² + 2² ✅
- Area 8: 2² + 2² ✅, Area 10: 1² + 3² ✅
- Area 3: kya koi a² + b² = 3? 0 + 3, 1 + 2… koi square nahi. Area 3 ka square dot grid pe nahi banta! ❌
Mumkin areas wahi hain jo do squares ke jod hon: 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20… (3, 6, 7, 11, 12 nahi). 🔐
sirf charumam.com readers ke liye 🔐
✏️ Solved Examples
Example 1 (Easy): 10 m seedhi deewar se 6 m door rakhi. Deewar pe kitni oonchi?
√(100 − 36) = √64 = 8 m.
WHY sahi hai? 6, 8, 10 triple ✅
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Example 2 (Medium): Rhombus ke diagonals 16 cm aur 30 cm. Side?
Legs 8 aur 15 → side = √(64 + 225) = √289 = 17 cm.
WHY sahi hai? (8, 15, 17) triple ✅
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Example 3 (NCERT style): Kamal ka phool paani se 2 haath upar, jhuk ke 6 haath door paani chhoota hai. Gehraai?
6² + x² = (x + 2)² → 36 = 4x + 4 → x = 8 haath.
WHY sahi hai? 6² + 8² = 100 = 10² (dandi 10) ✅
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Golu ka formula: “Sawaal mein 90° dhoondo, triangle apne aap dikh jaayega!” 🎉
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🏠 Real Life Mein
- 🪜 Seedhi aur deewar, 🪁 patang ki oonchaai, 📺 TV ka diagonal.
- 🗺️ Map pe seedhi doori: 3 km poorab, 4 km uttar → seedhi doori 5 km.
- 🏏 Cricket: Pitch ke kone se kone ka faasla, fielder ka shortcut throw.
- ⛵ Naav: Nadi paar karte waqt dhaara se tirchha raasta.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Diagonal (rectangle/square) | ✅ √(l² + b²) | Diagonal se 2 right triangles |
| Rhombus ki side | ✅ √((d₁/2)² + (d₂/2)²) | Diagonals 90° pe aadha karte hain |
| Equilateral ki height | ✅ √(a² − (a/2)²) | Altitude side ko aadha karti hai |
| Shape mein 90° nahi aur bana bhi nahi sakte | ❌ Seedhe theorem nahi | Pehle right triangle banao |
— 📌 charumam.com
❌→✅ Worked Mistake
Sawaal: Rhombus ke diagonals 24 aur 70. Side?
❌ Galat: “Side = √(24² + 70²) = √5476 = 74.”
Galti kya hai? Right triangle ke legs poore diagonals nahi, unke aadhe hain. 74 to diagonals se bhi bada ho gaya ❌
✅ Sahi:
- Legs 12 aur 35.
- Side = √(144 + 1225) = √1369 = 37.
Yaad rakho: Pehle chitra banao aur dekho right triangle ki asli sides kaun si hain.
sirf charumam.com pe milenge!
📝 10 MCQs
Q1. Kamal sawaal (1 upar, 3 door) mein gehraai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 3 to doori hai.
- (b) sahi: 9 = 2x + 1 → x = 4 ✅
- (c) galat: 5 dandi ki lambaai hai.
- (d) galat: Galat.
Q2. 5 cm square ka diagonal?
Solution dekho
Sahi jawab: (b)
- (a) galat: 2 × side nahi.
- (b) sahi: √50 ✅
- (c) galat: Area hai.
- (d) galat: 7² = 49 ≠ 50.
Q3. Rhombus diagonals 24 aur 70. Side?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Legs 12, 35 ✅
- (b) galat: Poore diagonals liye.
- (c) galat: Galat.
- (d) galat: Sirf aadha diagonal.
Q4. Hypotenuse hamesha sabse lambi side kyun?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Isliye c > a aur c > b ✅
- (b) galat: Saboot hai.
- (c) galat: Jagah se fark nahi.
- (d) galat: Hamesha hoti hai.
Q5. Side 6 wale equilateral triangle ki height?
Solution dekho
Sahi jawab: (b)
- (a) galat: 3 aadhi side hai.
- (b) sahi: 36 − 9 = 27 ✅
- (c) galat: Side ke barabar nahi.
- (d) galat: 36 + 9 jod diya.
Q6. Dot grid pe kaun sa area ka square NAHI ban sakta?
Solution dekho
Sahi jawab: (b)
- (a) galat: 1 + 1
- (b) sahi: Koi a² + b² = 3 nahi ✅
- (c) galat: 1 + 4
- (d) galat: 4 + 4
Q7. Kaun sa rectangle integer diagonal deta hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: √13.
- (b) sahi: 13 ✅
- (c) galat: √41.
- (d) galat: √5.
Q8. “Har triple primitive hai ya kisi primitive ka scaled roop.” Yeh…
Solution dekho
Sahi jawab: (a)
- (a) sahi: HCF se bhaag do → primitive milta hai ✅
- (b) galat: Hamesha sach.
- (c) galat: Hamesha.
- (d) galat: Saboot hai.
Q9. 7 aur 5 side ke squares ka farak kis square ke barabar?
Solution dekho
Sahi jawab: (b)
- (a) galat: 2² = 4, galat.
- (b) sahi: 49 − 25 = 24 ✅
- (c) galat: Jod hai.
- (d) galat: Galat.
Q10. 3 km poorab, phir 4 km uttar. Shuru se seedhi doori?
Solution dekho
Sahi jawab: (b)
- (a) galat: Raaste ki lambaai, seedhi nahi.
- (b) sahi: √(9 + 16) ✅
- (c) galat: Farak hai.
- (d) galat: Guna hai.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Real-life sawaal mein triangle kaise dhoondhein?
Pehle chitra banao. Jahan bhi 90° ho (deewar-zameen, uttar-poorab, rhombus diagonals), wahan right triangle hai.
2. Bhaskaracharya kaun the?
12vi sadi ke mahaan Bharatiya ganitagya. Līlāvatī aur Bījagaṇita unki prasiddh kitaabein hain.
3. Kamal wale sawaal mein x² kaise gaya?
(x + 1)² = x² + 2x + 1. Dono taraf x² hai, to ghatane pe hat jaata hai.
4. 9√3 kya hai?
√27 = √(9 × 3) = 3√3. Area = ½ × 6 × 3√3 = 9√3 ≈ 15.59.
5. Dot grid wale squares ka niyam kya hai?
Area = a² + b² (a, b poore numbers). Isliye 3, 6, 7 jaise areas nahi bante.
6. Rang wali paheli ka hint?
Saare label galat hain! Ek dibba kholo, baaki do apne aap tay ho jaayenge (HOTS dekho).
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy Rectangle 9 × 12 ka diagonal?
Solution dekho
15.
Q2. Easy Seedhi 13 m, deewar se 5 m. Oonchaai?
Solution dekho
12 m.
Q3. Easy 8 cm square ka diagonal (approx)?
Solution dekho
8√2 ≈ 11.31 cm.
Q4. Easy Rhombus diagonals 6 aur 8. Side?
Solution dekho
5.
Q5. NCERT style Integer side aur diagonal wale 5 rectangles batao.
Solution dekho
3 × 4, 6 × 8, 5 × 12, 8 × 15, 7 × 24.
Q6. NCERT style 7 aur 5 ke squares ke farak jitna square kaise banaoge?
Solution dekho
Hypotenuse 7, leg 5 ka right triangle; doosri leg pe square = 24.
Q7. NCERT style Dot grid pe area 2, 4, 5 ke square banao (kaise?).
Solution dekho
2: (1,1) wali side; 4: 2 × 2 seedha; 5: (1,2) wali side. Area 3 nahi banta.
Q8. Medium Equilateral triangle side 4. Height?
Solution dekho
√(16 − 4) = √12 ≈ 3.46.
Q9. Medium Kamal 1 haath upar, 5 haath door. Gehraai?
Solution dekho
25 = 2x + 1 → x = 12.
Q10. Medium Phone screen 6 cm × 8 cm. Diagonal?
Solution dekho
10 cm.
Q11. NCERT style Hypotenuse sabse lambi side hai, justify karo.
Solution dekho
c² = a² + b² > a² → c > a; isi tarah c > b.
Q12. NCERT style Kya har triple primitive ya scaled primitive hai?
Solution dekho
Haan: HCF f se bhaag do → primitive (a/f, b/f, c/f), aur asli = f × usse.
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🧠 HOTS (Sochne Wale Sawaal)
Q1. Rang ki paheli: teen dibbe (sirf laal, sirf neela, sirf hara), label RED, BLUE, GREEN, par koi label sahi nahi. Sirf ek dibba khol ke sab kaise pata karoge?
Solution dekho
“RED” wala kholo. Maano usme neela nikla. Ab “GREEN” wale mein hara nahi ho sakta (label galat) aur neela ho chuka, to usme laal. Bacha “BLUE” → hara. (Agar “RED” mein hara nikle, to isi tarah: “BLUE” → laal, “GREEN” → neela.)
Q2. Kamal dandi x + 1, doori d. Gehraai ka formula?
Solution dekho
d² + x² = (x + 1)² → x = (d² − 1)/2.
Q3. Ek square ki side 1 hai. Kya dot grid pe usse bilkul double area ka square ban sakta hai?
Solution dekho
Haan, area 2 = 1² + 1², tircha square.
Q4. 3 km poorab, 4 km uttar, phir 12 km upar (helicopter!). Shuru se seedhi doori?
Solution dekho
√(9 + 16) = 5; phir √(25 + 144) = 13 km.
sirf charumam.com pe milenge!
🏆 Olympiad Corner
Q1. Dot grid pe 1 se 20 tak kitne areas ke squares mumkin hain?
Solution dekho
1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20 → 12.
Q2. Equilateral triangle side 10. Area (approx)?
Solution dekho
Height = √75 ≈ 8.66. Area ≈ ½ × 10 × 8.66 = 43.3.
Q3. Rhombus ki side 13, ek diagonal 10. Doosra diagonal?
Solution dekho
Aadha = √(169 − 25) = 12 → diagonal 24.
Q4. Bamboo 18 haath ooncha, toot ke sira zameen pe 6 haath door gira. Kitni oonchaai pe toota?
Solution dekho
x² + 36 = (18 − x)² → 36 = 324 − 36x → x = 8 haath.
sirf charumam.com readers ke liye 🔐
💗 CharuMam Motivation
Bhaskaracharya ne ganit ko kavita aur kamal ke phoolon mein likha. Unke liye ganit sundarta tha, darr nahi. 🪷
Aapne poora chapter kar liya: double square se Fermat tak! Ab jahan bhi 90° dikhe, aapko ek chhupa hua triangle dikhega. Yahi ganit ki nazar hai. Badhaai ho! 🎓
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🟢 Master Recap
- Real-life mein pehle chitra banao, 90° dhoondo.
- Kamal: 3² + x² = (x + 1)² → x = 4.
- Diagonal = √(l² + b²); square ka diagonal = side × √2.
- Rhombus: legs = aadhe diagonals.
- Equilateral (side a): height = √(a² − (a/2)²); side 6 → area 9√3 ≈ 15.6.
- Dot grid square ka area = a² + b²; area 3 mumkin nahi.
- Har triple = primitive ya uska scaled roop.
— charumam.com 🎯
👉 Aage Kya?
🎉 Chapter 2 poora! Ab chapter hub se koi bhi post dobara dekho, ya agle chapter Proportional Reasoning-2 mein chalo!
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬