Class 10 Maths me jab Tribhuj (Triangles) chapter khulta hai to kai students ko lagta hai ki ye sirf “aakar dekhne” wala chapter hai. Asal me ye chapter board exam ke sabse zyada marks wale chapters me se hai, aur iske concepts aage Class 11-12 ki geometry aur trigonometry ki neev hain. tribhuj class 10 notes ki is post me aapko poora Chapter 6 milega: samroop tribhuj kya hote hain, unki kasautiyan, Thales pramey, kshetrafal ka niyam, aur Pythagoras theorem — sab kuch solved examples ke saath, simple Hindi me. Har example ko haath se check karke likha gaya hai, taaki aap bina dare inhe apni copy me utaar sakein.
Tribhuj Class 10 Notes — Samroop Tribhuj Kya Hote Hain
Do tribhuj samroop (similar) kehlate hain jab unke aakar ek jaise hon, lekin size alag ho sakta hai. Simple bhasha me: ek tribhuj ko bada ya chhota karke doosra tribhuj banaya ja sake, bina aakar badle. Samroop tribhujon ke liye do shartein hoti hain, aur dono ka poora hona zaroori hai:
- Unke corresponding kon barabar hon.
- Unki corresponding bhujayon ka anupaat barabar ho.
Misaal: agar tribhuj ABC aur tribhuj DEF samroop hain, to kon A = kon D, kon B = kon E, kon C = kon F, aur AB/DE = BC/EF = AC/DF. Is anupaat ko samroopta anupaat (scale factor) kehte hain.
Dhyan rakhne wali baat: sarvangsam (congruent) aur samroop me farq hai. Sarvangsam me size bhi barabar hota hai, samroop me sirf aakar barabar hota hai, size alag ho sakta hai. Har sarvangsam tribhuj samroop hota hai, lekin har samroop tribhuj sarvangsam nahi hota. Board exam me ye farq poochha jata hai — is line ko yaad rakh lijiye. tribhuj class 10 notes me ye farq har saal 1 mark ke sawal me aata hai.
Samroopta Ki Kasautiyan
Har baar teenon kon aur teenon bhujayein check karna mushkil hota hai, isliye teen kasautiyan hain — inme se koi ek poori ho to tribhuj samroop maan liye jaate hain:
AA Kasauti (Angle-Angle)
Agar ek tribhuj ke do kon doosre tribhuj ke do konon ke barabar hon to dono tribhuj samroop hote hain. Teesra kon apne aap barabar ho jata hai, kyunki teenon konon ka yog 180 degree hota hai.
Misaal: Tribhuj ABC me kon A = 50 degree aur kon B = 60 degree hai. Tribhuj DEF me kon D = 50 degree aur kon E = 60 degree hai. To kon C = 180 – 50 – 60 = 70 degree, aur kon F bhi 70 degree hoga. Isliye dono tribhuj AA kasauti se samroop hain.
SSS Kasauti (Side-Side-Side)
Agar ek tribhuj ki teenon bhujayein doosre tribhuj ki teenon bhujayon ke anupaat me hon to dono tribhuj samroop hote hain.
Misaal: Tribhuj ABC ki bhujayein 3 cm, 4 cm, 5 cm hain aur tribhuj DEF ki bhujayein 6 cm, 8 cm, 10 cm hain. Anupaat: 3/6 = 4/8 = 5/10 = 1/2. Anupaat barabar hai, isliye dono tribhuj SSS kasauti se samroop hain.
SAS Kasauti (Side-Angle-Side)
Agar ek tribhuj ki do bhujayein doosre tribhuj ki do bhujayon ke anupaat me hon, aur un bhujayon ke beech ka kon barabar ho, to dono tribhuj samroop hote hain.
Misaal: Tribhuj ABC me AB = 4 cm, AC = 6 cm aur unke beech ka kon A = 60 degree hai. Tribhuj DEF me DE = 8 cm, DF = 12 cm aur kon D = 60 degree hai. Anupaat: 4/8 = 6/12 = 1/2, aur beech ka kon barabar hai. Isliye dono tribhuj SAS kasauti se samroop hain.
Ek chetavni: sirf do bhujayon ka anupaat barabar hona kaafi nahi hai — beech ka kon bhi barabar hona chahiye. Agar kon barabar nahi hai to SAS kasauti nahi lagegi. Kai students yahin galti kar dete hain — wo sirf sides ka anupaat dekh kar samroop likh dete hain.
Thales Pramey (Basic Proportionality Theorem)
Ye is chapter ka sabse important pramey hai. Thales pramey kehta hai: agar kisi tribhuj ki ek bhujha ke samanantar ek rekha kheenchi jaye jo doosri do bhujayon ko kaat ti hai, to wo dono bhujayon ko ek hi anupaat me baant ti hai.
Maan lijiye tribhuj ABC me DE rekha BC ke samanantar hai, aur DE, AB ko D par aur AC ko E par kaat ti hai. To AD/DB = AE/EC hoga.
Solved Example 1 — Thales Pramey
Tribhuj ABC me DE, BC ke samanantar hai. AD = 3 cm, DB = 2 cm, AE = 4.5 cm hai. EC nikaaliye.
Hal: Thales pramey se, AD/DB = AE/EC. Yani 3/2 = 4.5/EC. Cross-multiply karne par: 3 x EC = 4.5 x 2 = 9. Isliye EC = 9/3 = 3 cm. Jawaab: EC = 3 cm.
Iska ulta bhi sach hai (converse): agar ek rekha tribhuj ki do bhujayon ko ek hi anupaat me baant ti hai to wo teesri bhujha ke samanantar hoti hai. Misaal: agar AD/DB = AE/EC hai to DE, BC ke samanantar hai.
Madhya-Bindu Pramey (Mid-Point Theorem)
Thales pramey ka ek khaas case: jo rekha do bhujayon ke madhya-binduon ko jodti hai, wo teesri bhujha ke samanantar hoti hai aur uski aadhi hoti hai. Misaal: agar D aur E, AB aur AC ke madhya-bindu hain to DE, BC ke samanantar hai aur DE = BC ka aadha hai. Ye result numericals me bahut kaam aata hai.
Samroop Tribhujon Ke Kshetrafal Ka Niyam
Do samroop tribhujon ke kshetrafalon ka anupaat unki corresponding bhujayon ke vargon ke anupaat ke barabar hota hai.
Misaal: Agar tribhuj ABC aur tribhuj DEF samroop hain aur unki corresponding bhujayon ka anupaat 3:5 hai, to unke kshetrafalon ka anupaat 3 ka varg : 5 ka varg = 9:25 hoga.
Solved Example 2 — Kshetrafal Ka Anupaat
Do samroop tribhujon ki corresponding bhujayein 3 cm aur 5 cm hain, aur chhote tribhuj ka kshetrafal 27 sq cm hai. Bade tribhuj ka kshetrafal nikaaliye.
Hal: Kshetrafalon ka anupaat = 3 ka varg : 5 ka varg = 9:25. Maan lijiye bade tribhuj ka kshetrafal x hai. To 27/x = 9/25. Cross-multiply: 9x = 27 x 25 = 675. Isliye x = 675/9 = 75 sq cm. Jawaab: 75 sq cm.
Isi tarah parimap (perimeter) ka anupaat corresponding bhujayon ke anupaat ke barabar hota hai — parimap ka varg nahi hota, ye farq yaad rakhiye. Kshetrafal me varg lagta hai, parimap me nahi.
Pythagoras Theorem
Samkon tribhuj me: karn ka varg = aadhaar ka varg + lamb ka varg. Agar tribhuj ABC me kon B samkon hai to AC ka varg = AB ka varg + BC ka varg. Ise aise likhte hain: c ka varg = a ka varg + b ka varg, jahan c karn hai.
Solved Example 3 — Pythagoras Se Bhujha Nikaalna
Ek samkon tribhuj ki do bhujayein 6 cm aur 8 cm hain, aur 10 cm wali bhujha karn hai. Check kijiye ki ye samkon tribhuj hai ya nahi.
Hal: 6 ka varg + 8 ka varg = 36 + 64 = 100. Aur 10 ka varg = 100. Dono barabar hain, isliye ye samkon tribhuj hai. Ye 6-8-10 wala triplet board exam me baar-baar aata hai — iske saath 3-4-5 aur 5-12-13 wale triplets bhi yaad rakh lijiye.
Solved Example 4 — Seedhi (Ladder) Wala Sawal
Ek 13 metre lambi seedhi deewar se lagi hai, aur uska neeche wala sira deewar se 5 metre door hai. Deewar par seedhi kitni oonchai tak pahunchegi?
Hal: Ye samkon tribhuj banata hai: karn = 13 m, aadhaar = 5 m, oonchai = h. Pythagoras se: h ka varg = 13 ka varg – 5 ka varg = 169 – 25 = 144. Isliye h = 144 ka vargmool = 12 metre. Jawaab: 12 metre.
Pythagoras Ka Vilom (Converse)
Agar kisi tribhuj me sabse badi bhujha ka varg baaki do bhujayon ke vargon ke yog ke barabar ho to wo tribhuj samkon hota hai, aur samkon sabse badi bhujha ke saamne hota hai.
Misaal: Tribhuj ki bhujayein 5 cm, 12 cm, 13 cm hain. Check: 5 ka varg + 12 ka varg = 25 + 144 = 169 = 13 ka varg. Barabar hai, isliye ye samkon tribhuj hai, aur samkon 13 cm wali bhujha ke saamne hai.
Samroopta Ke Rozmarra Ke Prayog
Solved Example 5 — Chhaya (Shadow) Wala Sawal
Ek 6 metre oonche khambe ki chhaya 4 metre hai. Usi waqt ek tower ki chhaya 28 metre hai. Tower ki oonchai nikaaliye.
Hal: Suraj ki kirnein samanantar hoti hain, isliye khamba aur tower, apni-apni chhaya ke saath samroop tribhuj banate hain. Anupaat: tower ki oonchai / khambe ki oonchai = tower ki chhaya / khambe ki chhaya. Yani h/6 = 28/4 = 7. Isliye h = 6 x 7 = 42 metre. Jawaab: 42 metre.
Solved Example 6 — Samroop Tribhuj Se Bhujha Nikaalna
Tribhuj ABC, tribhuj DEF ke samroop hai. AB = 4 cm, DE = 6 cm, BC = 6 cm hai. EF nikaaliye.
Hal: Samroop tribhujon me AB/DE = BC/EF. Yani 4/6 = 6/EF. Cross-multiply: 4 x EF = 36. Isliye EF = 36/4 = 9 cm. Jawaab: 9 cm.
Board Exam Ke Liye Important Baatein
- Thales pramey ka statement aur uska converse — dono rat lein, kyunki 2-3 marks ke sawal inhi se aate hain.
- Pythagoras theorem aur uska vilom — statement ke saath ek-ek example likhna seekhein.
- Kshetrafal ka niyam: “anupaat ka varg” — ye line bhoolne par poora numerical galat ho jata hai.
- Diagram hamesha saaf banayein aur corresponding parts par nishaan lagayein. Diagram ke bina geometry ke sawal me poore marks nahi milte. tribhuj class 10 notes ki taiyari me diagram practice ko aakhri din tak jaari rakhiye.
- “Samroop” aur “sarvangsam” ka farq ek line me likhna seekhein — ye 1 mark ka pakka sawal hai.
- Numerical solve karte waqt har step likhiye — sirf answer likhne par step marking nahi milti.
Aakhri Revision Checklist
Exam se ek din pehle is checklist ko dohra lein: samroop ki paribhasha, teenon kasautiyan (AA, SSS, SAS), Thales pramey aur uska converse, kshetrafal ka niyam, Pythagoras theorem aur uska vilom, aur 3-4-5, 6-8-10, 5-12-13 wale triplets. Har topic ka ek example bina dekhe solve karke dekhiye — jo example atak jaye, use dobara padhiye. Ye checklist poori ho jaye to Chapter 6 aapka pakka samajhiye.
Ek Nazar Me Chapter (Table Box)
| Topic | Mukhya Baat |
|---|---|
| Samroop tribhuj | Aakar ek jaise, size alag ho sakta hai; kon barabar, bhujayon ka anupaat barabar |
| AA kasauti | Do kon barabar hon to tribhuj samroop |
| SSS kasauti | Teenon bhujayon ka anupaat barabar ho to samroop |
| SAS kasauti | Do bhujayon ka anupaat + beech ka kon barabar ho to samroop |
| Thales pramey | Samanantar rekha do bhujayon ko ek hi anupaat me baant ti hai |
| Kshetrafal niyam | Kshetrafalon ka anupaat = bhujayon ke anupaat ka varg |
| Pythagoras theorem | Karn ka varg = aadhaar ka varg + lamb ka varg |
| Board/Exam | NCERT Class 10 Maths, Chapter 6 — Board Exam 2027 |
Official Links
- NCERT Class 10 Mathematics Textbook (Chapter 6 — Triangles): https://ncert.nic.in/textbook.php
- NCERT Official Website: https://ncert.nic.in
- CBSE Official Website (Syllabus aur Sample Papers): https://cbse.gov.in
FAQ
Sawaal: Samroop aur sarvangsam tribhuj me kya farq hai?
Jawaab: Samroop me sirf aakar ek jaisa hota hai, size alag ho sakta hai. Sarvangsam me aakar aur size dono ek jaise hote hain. Har sarvangsam tribhuj samroop hota hai, lekin har samroop tribhuj sarvangsam nahi hota.
Sawaal: Thales pramey kya hai?
Jawaab: Agar kisi tribhuj ki ek bhujha ke samanantar rekha kheenchi jaye jo doosri do bhujayon ko kaat ti hai, to wo dono bhujayon ko ek hi anupaat me baant ti hai. Misaal: AD/DB = AE/EC.
Sawaal: Pythagoras theorem kab lagta hai?
Jawaab: Sirf samkon tribhuj me. Karn ka varg = aadhaar ka varg + lamb ka varg. Agar tribhuj samkon nahi hai to ye theorem nahi lagega.
Sawaal: Kshetrafal ke anupaat me varg kyun lagta hai?
Jawaab: Kyunki kshetrafal do bhujayon ka guna hota hai (lambai x chaudai jaisa), isliye anupaat ka varg hota hai. Parimap me sirf ek bhujha ka anupaat hota hai, isliye usme varg nahi lagta.
Sawaal: Board exam me is chapter se kitne marks ke sawal aate hain?
Jawaab: Chapter 6 se aam taur par 5-8 marks ke sawal aate hain — ek chhota aur ek lamba sawal, kabhi-kabhi case study ke saath. Thales aur Pythagoras se sawal lagbhag har saal aate hain.
- Students Category: https://jagatindia.com/category/students/
- Arithmetic Progression Class 10 Notes Hindi: https://jagatindia.com/arithmetic-progression-class-10-notes-hindi/
- Statistics Class 10 Notes Hindi: https://jagatindia.com/statistics-class-10-notes-hindi/