✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 8 cm side ke square ka area?
- Square ke saare angles kitne degree ke hote hain?
- Isosceles right triangle ke do baaki angles kitne ke hote hain?
Answers dekho
- 64 cm²
- 90° har ek
- 45° aur 45° (kyunki 180 − 90 = 90, do barabar hisson mein)
Teeno sahi? Shaabaash! Pichhla post yaad nahi? Pehle Post 1 — Square Ko Double Kaise Karein? padh lijiye.
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🎬 Kahani: Priya Ka Rumaal
Art class mein Priya ke paas ek chaukor rumaal tha, 20 cm × 20 cm. Teacher ne kaha, “Iska aadha area wala chaukor design banana hai, par kainchi nahi chalegi, sirf modna (fold) hai!”
Rohan bola, “Side aadhi kar do, 10 cm!” Priya ne turant hisaab lagaaya: 10 × 10 = 100, jabki poora 20 × 20 = 400. Yeh to chauthaai (¼) hai, aadha nahi! 🤔
Phir Priya ne rumaal ke chaaron kone beech mein mod diye… aur beech mein ek chhota, ghooma hua chaukor ban gaya. Kya yeh sach mein aadha hai? Chalo saboot dhoondhte hain! 🧐
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📖 Concept: Midpoints Jodo, Aadha Area Pao
Step 1: Side aadhi karna kaam kyun nahi karta?
Side ko ½ karo to area ½ × ½ = ¼ ho jaata hai. Jaise double mein 4 guna ho gaya tha, waise hi aadhe mein chauthaai.
WHY? Area dono dishaon se banta hai. Dono taraf aadha, to kul chauthaai.
Step 2: Sides ke beech ke points (midpoints) jodo
Bade square ki chaaron sides ke midpoints P, Q, R, S lo aur jodo. Andar ek tircha square PQRS banta hai.
Ab beech se east-west aur north-south lines kheencho. Bada square 4 chhote squares mein bat gaya, aur har chhota square PQRS ki ek side se 2 barabar triangles mein kat gaya. Kul 8 triangles.
PQRS ke andar inme se 4 hain (5, 6, 7, 8), aur bahar 4 (kone wale 1, 2, 3, 4). 4 out of 8 = aadha! ✅
WHY yeh Post 1 jaisa hai? Bilkul wahi figure hai, bas ulta dekha! Wahan chhote square ke diagonal pe double square bana tha. Yahan bada square, andar wale ka double hai, to andar wala bade ka aadha.
Step 3: Kaagaz mod ke banao 📄
- Ek chaukor kaagaz lo. Use aadha-aadha mod ke khol do, taaki beech ka point (centre) aur sides ke midpoints dikh jaayein.
- Chaaron kone ek-ek karke centre tak mod do.
- Ab jo chaukor dikh raha hai (PQRS), uski creases sides ke midpoints se guzarti hain.
- Khol ke dekho: chaaron mude hue kone milkar andar wale square ko poora dhak lete hain, bina ek-doosre pe chadhe.
WHY aadha? Kone ke 4 triangles bilkul andar ke square ko dhak lete hain. To kone = andar wala. Dono milke poora kaagaz. Isliye andar wala = aadha.
Step 4: Kaise pakka ki PQRS square hi hai?
- Sides barabar: Kone ke chaaron triangles mein do sides “aadhi side” hain aur beech mein 90° ka kona. Chaaron congruent (SAS), to unki teesri sides PQ, QR, RS, SP barabar.
- Kone 90°: Har kone ka triangle isosceles right triangle hai, to uske baaki angles 45°–45°. P pe seedhi line: 45° + ∠P + 45° = 180°, to ∠P = 90°. Aise hi Q, R, S.
WHY dono check? Sirf sides barabar ho to rhombus bhi ho sakta hai. Square ke liye sides barabar + kone 90°, dono chahiye.
Step 5: Aadhe square ki side kitni?
Andar wale square ki side = kone ke chhote square ka diagonal. 20 cm ke rumaal mein chhota square 10 cm ka, to PQRS ki side = 10 cm ka diagonal ≈ 10 × 1.414 ≈ 14.14 cm. Check: 14.14 × 14.14 ≈ 200 = 400 ka aadha ✅
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🔐 Deep Concept: Double Aur Aadha, Ek Hi Sikke Ke Do Pehlu
Is figure mein teen squares chhupe hain: chhota (kone wala, 2 triangles), beech wala PQRS (4 triangles) aur bada (8 triangles). Area: 2 : 4 : 8, har ek pichhle ka double!
Side ke hisaab se: agar bade ki side 2 hai, to beech wale ki ≈ 1.414 (√2) aur chhote ki 1. Har kadam pe side √2 guna, area 2 guna.
Gehri baat: Aadhe area ke liye side ko 2 se nahi, √2 (≈1.414) se bhaag karna padta hai. 20 ÷ 1.414 ≈ 14.14 cm. Rumaal wala jawab hi aa gaya! 🔐
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✏️ Solved Examples
Example 1 (Easy): 12 cm side ka square hai. Midpoints jodkar bane square ka area?
Bada area = 144 cm². Andar wala = 144 ÷ 2 = 72 cm².
WHY sahi hai? 8 triangles mein se 4, yaani aadha ✅
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Example 2 (Medium): Ek square ki side aadhi karne pe area 25 cm² aaya. Asli square ka area kitna tha, aur aadha area kitna hota?
Side aadhi = area ¼. Asli = 4 × 25 = 100 cm². Aadha area = 50 cm².
WHY sahi hai? 25 to 100 ka chauthaai hai, aadha nahi. Yahi Rohan wali galti thi ✅
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Example 3 (NCERT style): Midpoint wale square PQRS ka area 18 cm² hai. Bade square ki side batao.
Bada = 2 × 18 = 36 cm². Side = √36 = 6 cm.
WHY sahi hai? 6 × 6 = 36, aur 36 ka aadha 18 ✅
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Priya ka formula: “Aadha chahiye? Kone beech mein modo, midpoints jodo!” 🎉
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🏠 Real Life Mein
- 🧣 Rumaal aur napkin folding: Kone beech mein modne se aadhe size ka square design banta hai.
- 🏠 Farsh ka design: Square tile ke midpoints jodkar bana diamond, tile ka aadha hota hai, do rang ka pattern.
- ✉️ Envelope: Chaukor kaagaz ke kone beech mein modo, aur bana envelope jaisa aakaar!
- 🎨 Rangoli: Square ke andar square ke andar square: har agla aadha area.
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🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Area aadha karna hai | ✅ Midpoints jodo (ya kone centre tak modo) | Andar wala square = aadha |
| Area chauthaai karna hai | ✅ Side aadhi karo | ½ × ½ = ¼ |
| Aadha chahiye aur side aadhi kar di | ❌ Galat | Area ¼ ho jaayega |
| Aadhe square ki side chahiye | ✅ Side ÷ √2 (≈ × 0.707) | √2 × √2 = 2 |
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❌→✅ Worked Mistake
Sawaal: 16 cm side ke square ka aadha area wala square. Uski side?
❌ Galat: “Aadha area, to aadhi side: 8 cm.”
Galti kya hai? 8 × 8 = 64 cm², par 16 × 16 = 256, jiska aadha 128 hai. 64 to chauthaai hai! ❌
✅ Sahi:
- Aadha area = 128 cm².
- Side = midpoint square ki side = 8 cm wale chhote square ka diagonal ≈ 8 × 1.414 ≈ 11.31 cm.
- Check: 11.31 × 11.31 ≈ 128 ✅
Yaad rakho: Area ½ ke liye side ÷ 2 nahi, side ÷ √2.
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📝 10 MCQs
Q1. Square ki side aadhi karne pe area?
Solution dekho
Sahi jawab: (b)
- (a) galat: Aadha tab hota jab sirf ek disha badalti.
- (b) sahi: ½ × ½ = ¼ ✅
- (c) galat: Side ghati hai, area badh nahi sakta.
- (d) galat: Side badli to area bhi badlega.
Q2. Square ki sides ke midpoints jodne se bane square ka area?
Solution dekho
Sahi jawab: (b)
- (a) galat: ¼ to side aadhi karne pe aata hai.
- (b) sahi: 8 mein se 4 triangles, yaani ½ ✅
- (c) galat: ¾ nahi, kone ke 4 triangles bahar hain.
- (d) galat: Andar wala chhota hai.
Q3. Midpoint figure mein center lines kheenchne pe bada square kitne barabar triangles mein bat-ta hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: 4 chhote squares hain, triangles nahi.
- (b) galat: 6 se baraabar baant nahi hota.
- (c) sahi: 4 squares × 2 triangles = 8 ✅
- (d) galat: 16 bahut zyada.
Q4. Kaagaz ke kone centre tak modne pe kone ke triangles…
Solution dekho
Sahi jawab: (a)
- (a) sahi: Bilkul fit, isliye andar wala = aadha ✅
- (b) galat: Poora dhakte hain.
- (c) galat: Centre tak modne pe bahar nahi jaate.
- (d) galat: Bina overlap ke fit hote hain.
Q5. PQRS square hai, yeh dikhane ke liye kya chahiye?
Solution dekho
Sahi jawab: (c)
- (a) galat: Rhombus mein bhi sides barabar hoti hain.
- (b) galat: Ek angle se poori shape tay nahi hoti.
- (c) sahi: Dono shartein = square ✅
- (d) galat: Diagonals ki baat alag se karni padti.
Q6. 10 cm side ke square ka aadha area?
Solution dekho
Sahi jawab: (b)
- (a) galat: 25 = chauthaai.
- (b) sahi: 100 ÷ 2 = 50 ✅
- (c) galat: 100 poora area hai.
- (d) galat: 5 to side ka aadha hai, area nahi.
Q7. Midpoint square ka area 32 cm² hai. Bade square ki side?
Solution dekho
Sahi jawab: (b)
- (a) galat: 4 × 4 = 16, galat.
- (b) sahi: 2 × 32 = 64, √64 = 8 ✅
- (c) galat: 16 × 16 = 256.
- (d) galat: 64 to area hai, side nahi.
Q8. Kone ke triangle (midpoint figure) ke baaki do angles?
Solution dekho
Sahi jawab: (b)
- (a) galat: Do sides barabar hain, to angles barabar honge.
- (b) sahi: Isosceles right triangle: 45°–45° ✅
- (c) galat: 60°–60° ke saath 90° = 210°, galat.
- (d) galat: 0° ka angle triangle mein nahi hota.
Q9. Aadhe area ke square ki side, asli side se lagbhag kitni?
Solution dekho
Sahi jawab: (b)
- (a) galat: 0.5 guna se area ¼ aata.
- (b) sahi: 1 ÷ √2 ≈ 0.707 ✅
- (c) galat: 0.25 se area 1/16 ho jaata.
- (d) galat: 1.41 guna to double area ke liye hai.
Q10. Square ke andar midpoint square, uske andar phir midpoint square. Teesra square pehle ka kitna?
Solution dekho
Sahi jawab: (b)
- (a) galat: ½ to doosra hai.
- (b) sahi: ½ × ½ = ¼ ✅
- (c) galat: ⅛ teen baar ke baad.
- (d) galat: ⅓ ka koi rishta nahi.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Midpoint kya hota hai?
Kisi line ke bilkul beech ka point, jo use do barabar hisson mein baante.
2. Andar wala square tircha kyun dikhta hai?
Kyunki uske kone bade square ki sides ke beech mein hain. Ghuma ke dekho to woh bhi seedha square hai, bas 45° ghooma hua.
3. Kya yeh trick rectangle pe chalegi?
Rectangle ke midpoints jodne se rhombus banta hai, aur uska area bhi rectangle ka aadha hota hai! Par woh square nahi hoga.
4. Aadhe square ki side exact kaise likhein?
Side ÷ √2, ya (side × √2) ÷ 2. Jaise 10 cm ke liye 5√2 ≈ 7.07 cm.
5. Kya bina modne ke aadha square bana sakte hain?
Haan, midpoints naap ke pencil se jodo. Modna sirf aasaan tareeka hai.
6. SAS kya hai?
Do sides aur unke beech ka angle barabar ho to do triangles congruent hote hain (Side-Angle-Side).
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy 14 cm square ka midpoint square, area?
Solution dekho
196 ÷ 2 = 98 cm².
Q2. Easy Side aadhi karne pe area kitne guna?
Solution dekho
¼ guna.
Q3. Easy Midpoint square ka area 50 cm² hai. Bade square ka area?
Solution dekho
100 cm².
Q4. Easy Ek square 9 cm ka. Side aadhi karke area?
Solution dekho
4.5 × 4.5 = 20.25 cm² (81 ka ¼).
Q5. Medium 20 cm rumaal ke andar teen baar midpoint square banaya. Sabse chhote ka area?
Solution dekho
400 → 200 → 100 → 50 cm².
Q6. Medium Midpoint figure mein ek triangle ka area 6 cm² hai. Bade square aur andar wale square ke area?
Solution dekho
Bada = 8 × 6 = 48 cm², andar = 4 × 6 = 24 cm².
Q7. Medium Rohan ne 30 cm square ki side aadhi ki. Usse kitna area kam mila, aadhe ke mukable?
Solution dekho
Mila 225, chahiye 450. 225 cm² kam.
Q8. NCERT style PQRS ke angle P = 90° kaise? Batao.
Solution dekho
Kone ke dono triangles ke P pe angles 45° hain. Seedhi line 180°: 180 − 45 − 45 = 90°.
Q9. NCERT style Kone wale 4 triangles congruent kyun hain?
Solution dekho
Har ek mein do sides = aadhi side, beech mein 90°. SAS se congruent.
Q10. Medium 16 cm square ke aadhe area wale square ki side (lagbhag)?
Solution dekho
16 ÷ 1.414 ≈ 11.31 cm.
Q11. Medium Ek square ka area 72 cm². Midpoint square ki side?
Solution dekho
Andar = 36 cm², side = 6 cm.
Q12. NCERT style Kya midpoint square ka midpoint square, bade square ki side ka aadha side wala square hai?
Solution dekho
Haan! Area ¼ hai, to side ½. Doosra midpoint square seedha (tircha nahi) bhi dikhta hai.
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🧠 HOTS (Sochne Wale Sawaal)
Q1. Aadha karna aur double karna, dono mein kaun si line common hai?
Solution dekho
Chhote square ka diagonal. Double square ki side = diagonal, aur aadhe square ki side = (bade ke kone wale chhote square ka) diagonal.
Q2. Ek rangoli mein 64 cm² ka square hai, andar midpoint square, phir andar ek aur. Kitne baar karein ki area 4 cm² se kam ho?
Solution dekho
64 → 32 → 16 → 8 → 4 → 2. 4 se kam (2) aane ke liye 5 baar.
Q3. Rectangle 8 × 6 ke midpoints jodo. Bani shape aur area?
Solution dekho
Rhombus, diagonals 8 aur 6. Area = rectangle ka aadha = 24.
Q4. Kya kisi square ke andar aisa square ban sakta hai jiska area bilkul ⅓ ho, sirf modne se?
Solution dekho
Midpoint wali trick sirf ½, ¼, ⅛… deti hai (2 ki powers). ⅓ ke liye theorem wali construction chahiye (Post 5 dekho).
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🏆 Olympiad Corner
Q1. Square ki side 10 cm. Midpoints jodkar square, phir uske midpoints jodkar square, kul 4 baar. Aakhri square ki side?
Solution dekho
Area 100 → 50 → 25 → 12.5 → 6.25. Side = √6.25 = 2.5 cm.
Q2. Ek square ke andar midpoint square bana. Bahar ke 4 triangles ka kul area 32 cm² hai. Bade square ki side?
Solution dekho
Bahar = andar = aadha. Bada = 64 cm². Side = 8 cm.
Q3. Square ABCD (side 2). P, Q, R, S midpoints. PQ ki lambaai ka square kitna hai?
Solution dekho
PQRS ka area = 2, to PQ² = 2 (yaani PQ = √2).
Q4. Kya midpoint square ki perimeter, bade square ki perimeter ka aadha hai?
Solution dekho
Nahi! Side ≈ 0.707 guna hoti hai, to perimeter bhi ≈ 0.707 guna. Area aadha, perimeter aadha nahi.
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💗 CharuMam Motivation
Priya ke paas kainchi nahi thi, sirf ek kaagaz aur dimaag. Phir bhi usne saboot ke saath aadha square bana diya! 🧠
Kabhi lagta hai saadhan kam hain. Par sochne ka sahi tareeka ho to modne jaisi chhoti cheez se bhi bada ganit nikalta hai. Jo hai, usi se shuru karo! 💪
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🟢 Master Recap
- Side ½ → area ¼. Aadhe area ke liye side aadhi mat karo.
- Sides ke midpoints jodo → andar ka square = bade ka aadha.
- Saboot: bada = 8 triangles, andar = 4 triangles.
- Kaagaz: kone centre tak modo, kone andar wale ko poora dhakte hain.
- PQRS square hai: sides barabar (SAS) + har kona 90° (180 − 45 − 45).
- Aadhe square ki side = side ÷ √2 ≈ side × 0.707.
— charumam.com 🎯
👉 Aage Kya?
Double aur aadhe, dono mein ek rahasyamay number aaya: √2. Agle post mein dekhenge √2 kitna hai, uska decimal kabhi khatam kyun nahi hota, aur woh fraction kyun nahi ban sakta! 🔢
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬