Vritt Class 10 Notes – Circle Chapter Se Darne Ki Zaroorat Nahi
Board exam me geometry ke chapters se kai students ko dar lagta hai, aur Vritt unme se ek hai. Lekin sach ye hai ki NCERT ka Chapter 10, Vritt, sabse chhota aur sabse scoring chapter hai. Isme sirf 2 pramey hain, kuch paribhashayen hain aur wahi sawal ghoom-ghoom kar aate hain. Vritt class 10 notes is article me hum isi chapter ko itna simple bana denge ki aapko ye chapter sabse aasan lagne lagega.
Ye notes NCERT Class 10 Mathematics ke Chapter 10 par aadharit hain. Isme sparsh rekha ki paribhasha, kisi bindu se sparsh rekhaon ki sankhya, dono pramey unke proof ke saath, solved examples aur practice questions with answers diye gaye hain. Har example ko haath se check kiya gaya hai, taaki aap bina dare inhe copy karke samajh sakein.
Vritt Class 10 Notes – Ek Nazar Me Chapter
| Baat | Jaankari |
|---|---|
| Topic | Vritt (Circles) |
| Class | Class 10 Mathematics |
| NCERT Chapter | Chapter 10 |
| Isme kya hai | Sparsh rekha, uski sankhya, 2 pramey, solved examples, practice questions |
| Kis exam ke liye | CBSE/State Board Exam 2027 |
| Marks ka andaza | 3-5 marks, MCQ + short + numerical |
Vritt Class 10 Notes – Zaroori Paribhashayen
Vritt, yaani circle, un sabhi binduon ka samuh hai jo ek sthir bindu se ek nishchit doori par hote hain. Sthir bindu ko kendra kehte hain aur nishchit doori ko trijya, yaani radius kehte hain. Kendra se hokar jaane wali aur vritt ke do binduon ko jodne wali rekha ko vyas, yaani diameter kehte hain, jo trijya ka dugna hota hai.
Vritt ke do binduon ko jodne wali rekha ko jeeva, yaani chord kehte hain. Jo rekha vritt ko do alag-alag binduon par kaat ti hai, use chhedak rekha, yaani secant kehte hain. Aur jo rekha vritt ko sirf ek hi bindu par chhoo kar nikal jaati hai, use sparsh rekha, yaani tangent kehte hain. Jis bindu par sparsh rekha vritt ko chhooti hai, use sparsh bindu, yaani point of contact kehte hain.
Ek aur zaroori baat: vritt ke andar ka hissa, vritt par ke bindu aur vritt ke bahar ka hissa, ye teen alag kshetra hote hain. Sparsh rekhaon ki sankhya isi par depend karti hai ki bindu kahan hai, jo agle section me samajh aayega.
Vritt Class 10 Notes – Kisi Bindu Se Sparsh Rekhaon Ki Sankhya
Kisi bindu se vritt par kitni sparsh rekhaen kheenchi ja sakti hain, ye us bindu ki sthiti par depend karta hai. Teen cases hote hain, aur board exam me ye sawaal seedha poochha jaata hai.
Pehla case: agar bindu vritt ke andar hai, to us bindu se vritt par koi bhi sparsh rekha nahi kheenchi ja sakti. Sankhya zero hai. Doosra case: agar bindu vritt par hi hai, to us bindu se sirf ek hi sparsh rekha kheenchi ja sakti hai. Teesra case: agar bindu vritt ke bahar hai, to us bindu se vritt par exactly do sparsh rekhaen kheenchi ja sakti hain.
Ye teeno cases diagram ke saath samajhna bahut zaroori hai, kyunki 1 mark ke sawaal aksar isi par aate hain. Yaad rakhne ka aasan tareeka ye hai: andar se zero, upar se ek, bahar se do. Is line ko yaad rakhne se ye sawaal kabhi galat nahi hoga.
Vritt Class 10 Notes – Pramey 1, Sparsh Rekha Trijya Par Lamb Hoti Hai
Pramey 1 kehta hai: vritt ke kisi bindu par sparsh rekha, sparsh bindu se hokar jaane wali trijya par lamb hoti hai. Matlab agar O kendra hai, P sparsh bindu hai aur l sparsh rekha hai, to OP aur l ke beech 90 degree ka kon banta hai.
Iska proof samajhna aasan hai. Maan lein ki OP trijya hai aur l sparsh rekha hai. Agar OP, l par lamb nahi hai, to l par O se koi aur lamb daala ja sakta hai, jo l ko kisi bindu Q par milega. Lekin O se l ki sabse chhoti doori hi lamb hoti hai, aur koi bhi doosri doori trijya se badi hogi, isliye Q vritt ke bahar hoga. Lekin l vritt ko sirf P par chhooti hai, isliye aisa nahi ho sakta. Isliye OP hi l par lamb hai.
Is pramey ka sabse bada fayda ye hai ki isse har tangent wale sawaal me ek samkon tribhuj ban jaata hai. Jab bhi sparsh rekha aur trijya milein, to samkon tribhuj bana kar Pythagoras pramey lagayein, aur sawaal aasan ho jaata hai. Board exam ke lagbhag saare numerical isi ek idea par based hote hain.
Vritt Class 10 Notes – Pramey 2, Bahari Bindu Se Dono Sparsh Rekhaen Barabar Hoti Hain
Pramey 2 kehta hai: kisi bahari bindu se vritt par kheenchi gayi do sparsh rekhaon ki lambaiyan barabar hoti hain. Matlab agar P bahari bindu hai aur PA aur PB do sparsh rekhaen hain, to PA = PB hoga.
Iska proof bhi samkon tribhuj se hota hai. O kendra hai, OA aur OB trijyaen hain, aur OP dono tribhujon ki common bhuj hai. Tribhuj OAP aur OBP me, OA = OB (ek hi vritt ki trijyaen), kon OAP = kon OBP = 90 degree (pramey 1 se), aur OP common hai. Isliye RHS sarvangsamta se dono tribhuj sarvangsam hain, aur isliye PA = PB. Saath me ye bhi milta hai ki OP, kon APB aur kon AOB ko bhi do barabar hisson me baant ta hai.
Is pramey se ek aur zaroori baat milti hai. Chaturbhuj OAPB me, kon OAP + kon OBP = 180 degree hai, isliye kon AOB + kon APB = 180 degree bhi hota hai. Matlab kendra par bana kon aur bahari bindu par bana kon, dono ka yog 180 degree hota hai. Ye result bhi board exam me seedha poochha jaata hai.
Vritt Class 10 Notes – Solved Examples, Haath Se Check Kiye Gaye
Example 1: Ek vritt ki trijya 5 cm hai aur kendra se ek bahari bindu ki doori 13 cm hai. Sparsh rekha ki lambai nikaaliye. Hal: samkon tribhuj me, sparsh rekha ki lambai = root (13 ka square – 5 ka square) = root (169 – 25) = root 144 = 12 cm. Uttar 12 cm hai. Ye 5-12-13 ka famous triplet hai, ise yaad rakhne se waqt bachta hai.
Example 2: Ek bahari bindu P se vritt par do sparsh rekhaen PA aur PB kheenchi gayi hain. Agar kon APB = 60 degree hai, to PA aur AB ka sambandh kya hoga? Hal: PA = PB (pramey 2 se) aur beech ka kon 60 degree hai, isliye tribhuj APB samabahu hoga, kyunki do barabar bhujon ke beech 60 degree ka kon hone par teesri bhuj bhi barabar ho jaati hai. Isliye AB = PA. Agar PA = 6 cm hai, to AB bhi 6 cm hoga.
Example 3: Do sankendriya vritt hain, jinki trijyaen 5 cm aur 13 cm hain. Bade vritt ki wo jeeva jo chhote vritt ko sparsh karti hai, uski lambai nikaaliye. Hal: kendra se jeeva par lamb daalne par, aadhi jeeva = root (13 ka square – 5 ka square) = root (169 – 25) = 12 cm. Isliye poori jeeva = 2 x 12 = 24 cm. Uttar 24 cm hai. Ye NCERT ke exercise ka sabse famous sawaal hai.
Example 4: Ek vritt ka kendra O hai aur bahari bindu P se do sparsh rekhaen PA aur PB kheenchi gayi hain. Agar kon AOB = 120 degree hai, to kon APB kitna hoga? Hal: chaturbhuj OAPB me, kon OAP = 90 degree, kon OBP = 90 degree, kon AOB = 120 degree. Charon konon ka yog 360 degree hota hai, isliye kon APB = 360 – 90 – 90 – 120 = 60 degree. Uttar 60 degree hai.
Example 5: Ek vritt ki trijya 6 cm hai aur kendra se bahari bindu ki doori 10 cm hai. Sparsh rekha ki lambai nikaaliye. Hal: sparsh rekha = root (10 ka square – 6 ka square) = root (100 – 36) = root 64 = 8 cm. Uttar 8 cm hai. Ye 6-8-10 ka triplet hai, jo 3-4-5 ka dugna hai.
Vritt Class 10 Notes – Practice Questions With Answers
Question 1: Vritt ki trijya 8 cm hai aur kendra se bahari bindu ki doori 17 cm hai. Sparsh rekha ki lambai kya hogi? Answer: root (289 – 64) = root 225 = 15 cm.
Question 2: Sparsh rekha ki lambai 24 cm hai aur trijya 7 cm hai. Kendra se bahari bindu ki doori kitni hogi? Answer: root (576 + 49) = root 625 = 25 cm.
Question 3: Ek bahari bindu se do sparsh rekhaen kheenchi gayi hain aur unke beech ka kon 80 degree hai. Kendra par bana kon kitna hoga? Answer: 180 – 80 = 100 degree.
Question 4: PA aur PB do sparsh rekhaen hain, OA = 5 cm aur AP = 12 cm hai. OP ki lambai kya hogi? Answer: root (25 + 144) = root 169 = 13 cm.
Question 5: Vritt ke andar sthit bindu se kitni sparsh rekhaen kheenchi ja sakti hain? Answer: zero, koi bhi nahi.
In practice questions ko bina answer dekhe solve karein, phir answer se milaan karein. Jo sawaal galat ho, uska concept wapas padhein. Board exam me isi pattern ke sawaal aate hain, sirf numbers badal jaate hain.
Vritt Class 10 Notes – Board Exam Ke Liye Revision Points
Revision ke liye ye points yaad rakhein. Pehla, sparsh rekha ki paribhasha: jo rekha vritt ko sirf ek bindu par chhooe. Doosra, sparsh rekhaon ki sankhya: andar se zero, vritt par se ek, bahar se do. Teesra, pramey 1: sparsh rekha, sparsh bindu se jaane wali trijya par lamb hoti hai. Chautha, pramey 2: bahari bindu se dono sparsh rekhaen barabar hoti hain.
Paanchva, har tangent wale sawaal me samkon tribhuj banayein aur Pythagoras pramey lagayein. Chhatha, chaturbhuj OAPB me kon AOB + kon APB = 180 degree hota hai. Saatvaan, sankendriya vritt wale sawaal me bade vritt ki jeeva jo chhote ko sparsh karti hai, uski aadhi lambai Pythagoras se nikal kar dugni karein. In saat points se chapter ka 90% cover ho jaata hai.
Vritt Class 10 Notes – Diagram Banane Ka Sahi Tareeka
Geometry ke sawaalon me diagram aadha answer hota hai, aur board exam me diagram ke marks bhi milte hain. Vritt class 10 notes ke hisaab se diagram banate waqt pehle compass se saaf vritt banayein aur kendra ko O se mark karein. Sparsh rekha ko scale se seedhi rekha me banayein aur sparsh bindu ko P se mark karein. Trijya OP ko bhi scale se jodein, taaki samkon ka nishaan saaf dikhe.
Diagram me har bindu ka naam likhna zaroori hai, kyunki bina naam ke diagram ke marks kat jaate hain. Konon ko bhi mark karein, jaise samkon par chhota square ka nishaan lagayein. Agar sawaal me do sparsh rekhaen hain, to dono ko PA aur PB naam dein aur kendra se dono trijyaen OA aur OB bhi banayein. Saaf aur naam wale diagram se examiner ko aapka tareeka samajh aata hai aur step marking me bhi fayda milta hai. Vritt class 10 notes ki taiyaari me roz ek diagram banane ki aadat daalne se exam me waqt bhi bachta hai aur galti bhi kam hoti hai.
Vritt Class 10 Notes – Aksar Poochhe Jaane Wale Sawaal
Sparsh rekha aur chhedak rekha me kya farq hai?
Sparsh rekha vritt ko sirf ek bindu par chhooti hai, jabki chhedak rekha vritt ko do alag-alag binduon par kaat ti hai. Diagram me ye farq sabse aasani se samajh aata hai, isliye dono ka diagram banakar dekhna chahiye.
Vritt ke bahar sthit bindu se kitni sparsh rekhaen kheenchi ja sakti hain?
Exactly do sparsh rekhaen kheenchi ja sakti hain, aur pramey 2 ke hisaab se dono ki lambai barabar hoti hai. Vritt ke andar wale bindu se zero aur vritt par wale bindu se sirf ek sparsh rekha kheenchi ja sakti hai.
Sparsh rekha trijya par lamb kyun hoti hai?
Ye pramey 1 hai. Proof me dikhaya jaata hai ki agar trijya lamb nahi hoti, to sparsh rekha vritt ko ek se zyada bindu par kaat ti, jo sparsh rekha ki paribhasha ke khilaaf hai. Isliye sparsh bindu par trijya aur sparsh rekha ke beech 90 degree ka kon banta hai.
Tangent wale numerical solve karne ka sabse aasan tareeka kya hai?
Sparsh bindu, kendra aur bahari bindu ko milakar samkon tribhuj banayein. Usme Pythagoras pramey lagayein: sparsh rekha ka square + trijya ka square = kendra se bindu ki doori ka square. Is ek formula se lagbhag saare numerical hal ho jaate hain.
Kya chapter 10 se board exam me kitne marks ke sawaal aate hain?
Is chapter se aam taur par 3-5 marks ke sawaal aate hain, jisme 1 mark ke MCQ, 2 marks ke short answer aur 3 marks ke numerical shaamil hote hain. Chhota chapter hone ki wajah se ise poora taiyaar karna sabse aasan hai.
Aur bhi padhein: Tribhuj Class 10 Notes Chapter 6 aur Bahupad Class 10 Notes Chapter 2.
Official Links
- NCERT Official Website: https://ncert.nic.in
- NCERT Class 10 Mathematics Textbook: https://ncert.nic.in/textbook.php
- CBSE Official Website: https://www.cbse.gov.in
Aakhri Baat
Vritt ka chapter chhota hai, lekin iski taiyaari me laparwahi mehngi pad sakti hai, kyunki iske sawaal pakke aate hain. Dono pramey, unke proof aur Pythagoras wale numerical ko itna pakka kar lein ki koi bhi number badal de, to bhi sawaal hal ho jaaye. Practice questions ko khud solve karein, diagram banayein aur revision points ko exam se pehle ek baar zaroor padhein. Ye chapter aapke liye free ke marks jaisa hai, bas ise chhodna mat.