Nirdeshank Jyamiti Class 10 Notes: Graph Paper Se Dar Kyun Lagta Hai
Graph paper dekhte hi kai students ke dimag me x-axis, y-axis aur coordinates ka jaal ghoomne lagta hai. Kaun sa point kahan hai, doori kaise nikaalein, beech ka point kaise dhoondhein — ye sab ek saath dekh kar chapter mushkil lagta hai. Lekin sach ye hai ki ye chapter sirf 3-4 formula ka hai, aur ek baar ye samajh aa jaye ki coordinate ka matlab kya hai, to poora chapter aasan ho jaata hai. Nirdeshank jyamiti class 10 notes ka ye guide isi samajh par bana hai.
Nirdeshank jyamiti ka matlab hai jyamiti ko sankhyaon ki madad se samajhna. Har point ki ek pehchaan hoti hai — (x, y) — jisme x batata hai ki point y-axis se kitni doori par hai aur y batata hai ki point x-axis se kitni doori par hai. Sochiye ki aapko kisi sheher ka naksha diya gaya hai: pehli sankhya batati hai ki ghar sadak par kitna aage hai, doosri batati hai ki kitna upar hai. Bas yahi coordinate system hai. Is chapter me aap doori (distance), vibhajan (section), madhyabindu (midpoint) aur tribhuj ka kshetrafal seekhenge.
| Topic | Formula | Kya batata hai | ||
|---|---|---|---|---|
| Doori (Distance) | √[(x2 − x1)² + (y2 − y1)²] | Do points ke beech ki seedhi doori | ||
| Madhyabindu (Midpoint) | ((x1 + x2)/2, (y1 + y2)/2) | Do points ke beech ka point | ||
| Vibhajan (Section) | ((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n)) | Rekha ko m:n me baantne wala point | ||
| Tribhuj ka kshetrafal | ½\ | x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)\ | Teen points se bane tribhuj ka area | |
| Collinearity | Kshetrafal = 0 | Teen points ek hi rekha par hain |
Nirdeshank Jyamiti Class 10 Notes: Coordinate System Ki Buniyaad
Nirdeshank jyamiti class 10 notes ki shuruaat coordinate system se hoti hai. X-axis ek horizontal rekha hai aur y-axis ek vertical rekha hai; dono jahan milti hain, use moolbindu (origin) kehte hain, jiska coordinate (0, 0) hota hai. X-axis ke upar y positive hota hai aur neeche y negative; y-axis ke daayein x positive hota hai aur baayein x negative. Is tarah poora plane 4 chaturthansh (quadrants) me bat jaata hai: pehle me (+, +), doosre me (−, +), teesre me (−, −) aur chauthe me (+, −).
Ek aam galti yahan hoti hai: x aur y ka kram ulta likh dena. Hamesha pehle x, phir y likha jaata hai — (x, y) ka matlab hai “x ki disha me itna, phir y ki disha me itna”. Agar aap (3, 4) ki jagah (4, 3) likh dein to point hi badal jayega. Nirdeshank jyamiti class 10 notes ki taiyaari me isi chhoti galti se poora sawaal galat ho jaata hai, isliye coordinate likhte samay kram ka dhyan rakhein.
X-axis par kisi bhi point ka y-coordinate 0 hota hai, aur y-axis par kisi bhi point ka x-coordinate 0 hota hai. Ye baat kai sawaalon me kaam aati hai — jaise jab poocha jaye ki x-axis par wo point dhoondhein jo do points se barabar doori par hai, to us point ko (x, 0) maan kar sawaal hal kiya jaata hai. Aage ke solved examples me aap ye tareeka dekhenge.
Distance Formula — Do Points Ke Beech Ki Doori
Distance formula Pythagoras pramey se hi aata hai, isliye ise ratne ki zaroorat nahi hai, samajhne ki hai. Nirdeshank jyamiti class 10 notes me ye formula sabse pehle aata hai aur sabse zyada istemal hota hai. Do points P(x1, y1) aur Q(x2, y2) ke beech ki doori hoti hai: PQ = √[(x2 − x1)² + (y2 − y1)²]. Iska matlab hai: x-coordinates ka antar lo, uska varg karo; y-coordinates ka antar lo, uska varg karo; dono ko jodo aur vargmool nikaalo. Bas.
Solved Example 1: Points (3, 4) aur (7, 10) ke beech ki doori nikaalein. X ka antar = 7 − 3 = 4, y ka antar = 10 − 4 = 6. Doori = √(4² + 6²) = √(16 + 36) = √52 = 2√13 units. Yahan √52 ko 2√13 likhna hi sahi uttar hai — decimal me badalne ki zaroorat nahi hai, aur board me surd roop hi maana jaata hai.
Solved Example 2: X-axis par wo point dhoondhein jo (2, −5) aur (−2, 9) se barabar doori par hai. X-axis par point ko (x, 0) maan lein. Pehle point se doori ka varg = (x − 2)² + (0 + 5)² = (x − 2)² + 25. Doosre point se doori ka varg = (x + 2)² + (0 − 9)² = (x + 2)² + 81.
Dono barabar rakhne par: (x − 2)² + 25 = (x + 2)² + 81. Vistar karne par: x² − 4x + 4 + 25 = x² + 4x + 4 + 81, matlab −4x + 29 = 4x + 85, isliye −8x = 56 aur x = −7. Answer: (−7, 0). Jaanch: (−7, 0) se (2, −5) ki doori = √(81 + 25) = √106, aur (−2, 9) se doori = √(25 + 81) = √106. Dono barabar — uttar sahi hai.
Ek practical tip: doori nikaalte samay antar ka chihn (plus/minus) matter nahi karta, kyunki varg karne par minus bhi plus ho jaata hai. Lekin antar galat nikaalna bahut mehnga padta hai, isliye (x2 − x1) aur (y2 − y1) dhyan se nikaalein.
Section Formula — Rekha Ko Baantne Wala Point
Section formula tab kaam aata hai jab kisi rekha ko kisi anupaat me baantna ho. Nirdeshank jyamiti class 10 notes me ye doosra sabse zaroori formula hai. Agar point P, points A(x1, y1) aur B(x2, y2) ko milane wali rekha ko m:n ke anupaat me baantta hai, to P ka coordinate hota hai: ((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n)). Yaad rakhne ka aasan tareeka ye hai: jis hisse ka anupaat m hai, us taraf wale point ka coordinate n se guna hota hai — matlab ulta guna hota hai. Ye “ulta” hi sabse zyada galti ki wajah banta hai, isliye formula likhte samay dhyan dein.
Solved Example 3: Points (2, 3) aur (6, 7) ko milane wali rekha ko 1:2 ke anupaat me baantne wala point nikaalein. Yahan m = 1, n = 2, (x1, y1) = (2, 3), (x2, y2) = (6, 7). X-coordinate = (1×6 + 2×2)/(1 + 2) = (6 + 4)/3 = 10/3. Y-coordinate = (1×7 + 2×3)/3 = (7 + 6)/3 = 13/3. Answer: (10/3, 13/3). Jaanch ka tareeka: ye point dono points ke beech hi hona chahiye — 10/3 lagbhag 3.33 hai, jo 2 aur 6 ke beech hai, aur 13/3 lagbhag 4.33 hai, jo 3 aur 7 ke beech hai. Sahi hai.
Midpoint formula section formula ka hi khaas roop hai — jab anupaat 1:1 ho. Tab formula ban jaata hai: ((x1 + x2)/2, (y1 + y2)/2). Solved Example 4: Points (−2, 5) aur (4, −3) ka madhyabindu = ((−2 + 4)/2, (5 + (−3))/2) = (2/2, 2/2) = (1, 1). Simple hai, lekin sign ki galti yahan bhi hoti hai — negative sankhyaon ko jodte samay dhyan dein.
Trisection ka matlab hai rekha ko 3 barabar hisson me baantna. Solved Example 5: Points A(2, 1) aur B(8, 7) ko 3 barabar hisson me baantne wale points nikaalein. Pehla point A se 1:2 ke anupaat me hai: ((1×8 + 2×2)/3, (1×7 + 2×1)/3) = (12/3, 9/3) = (4, 3). Doosra point A se 2:1 ke anupaat me hai: ((2×8 + 1×2)/3, (2×7 + 1×1)/3) = (18/3, 15/3) = (6, 5). Answer: (4, 3) aur (6, 5). Dono points A aur B ke beech me hain aur barabar doori par hain — yahi trisection ki pehchaan hai.
Tribhuj Ka Kshetrafal Aur Collinearity
Teen points se bane tribhuj ka kshetrafal nikaalne ka formula hai: Area = ½|x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)|. Modulus (khade daand) isliye lagaya jaata hai kyunki kshetrafal kabhi negative nahi hota — agar andar ki value negative aaye to uska positive maan liya jaata hai.
Solved Example 6: Points (0, 0), (4, 0) aur (0, 3) se bane tribhuj ka kshetrafal. Formula me rakhne par: Area = ½|0×(0 − 3) + 4×(3 − 0) + 0×(0 − 0)| = ½|0 + 12 + 0| = 6 sq units. Samajh se bhi ye sahi hai: ye ek samkon tribhuj hai jiska aadhaar 4 aur unchai 3 hai, to kshetrafal = ½ × 4 × 3 = 6. Dono tareeke ek hi jawab dete hain.
Solved Example 7: Kya points (1, 2), (3, 4) aur (5, 6) ek hi rekha par hain (collinear)? Kshetrafal nikaalein: Area = ½|1×(4 − 6) + 3×(6 − 2) + 5×(2 − 4)| = ½|−2 + 12 − 10| = ½ × 0 = 0. Kshetrafal 0 aane ka matlab hai ki teenon points ek hi rekha par hain — ye collinearity ki shart hai. Exam me ye sawaal aksar aata hai, aur iska jawab sirf “area = 0” kehna kaafi nahi hai — poora calculation dikhana zaroori hai.
Centroid (kendrak) bhi isi chapter ka ek pyaara topic hai. Tribhuj ke teenon sheershon ka ausat hi centroid hota hai: ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3). Solved Example 8: Sheersh (2, 3), (4, 7) aur (6, 1) wale tribhuj ka centroid = ((2 + 4 + 6)/3, (3 + 7 + 1)/3) = (12/3, 11/3) = (4, 11/3). Centroid wahi point hai jahan tribhuj ki teenon madhyikayein milti hain — ye baat figure banane me bhi kaam aati hai.
Practice Questions — Khud Karke Dekhein
Ab kuch sawaal khud hal karein. Nirdeshank jyamiti class 10 notes ki asli taiyaari practice se hi hoti hai. Jawab neeche diye gaye hain, lekin pehle khud koshish karein — bina koshish ke jawab dekhna samay barbaad karna hai.
Practice 1: Points (5, 7) aur (1, 4) ke beech ki doori nikaalein. Practice 2: Points (1, 5), (2, 3) aur (−2, −1) se bane tribhuj ka kshetrafal nikaalein. Practice 3: Points (3, 2) aur (9, 8) ka madhyabindu nikaalein. Practice 4: Kya points (2, 3), (4, 5) aur (6, 7) collinear hain? Practice 5: Y-axis par wo point dhoondhein jo (5, −2) aur (−3, 2) se barabar doori par hai.
Jawab: Practice 1: Doori = √[(5 − 1)² + (7 − 4)²] = √(16 + 9) = √25 = 5 units. Practice 2: Area = ½|1×(3 + 1) + 2×(−1 − 5) + (−2)×(5 − 3)| = ½|4 − 12 − 4| = 6 sq units. Practice 3: Midpoint = ((3 + 9)/2, (2 + 8)/2) = (6, 5).
Practice 4: Area = ½|2×(5 − 7) + 4×(7 − 3) + 6×(3 − 5)| = ½|−4 + 16 − 12| = 0, isliye haan, collinear hain. Practice 5: Y-axis par point (0, y) maan kar: (0 − 5)² + (y + 2)² = (0 + 3)² + (y − 2)², matlab 25 + y² + 4y + 4 = 9 + y² − 4y + 4, isliye 8y = −16 aur y = −2. Answer: (0, −2).
Board Exam Tips Aur Aam Galtiyan
Board exam me is chapter se aam taur par 3-4 marks ke sawaal aate hain, kabhi 5 marks ka bhi. Nirdeshank jyamiti class 10 notes ki taiyaari me sabse zyada marks isi baat par milte hain ki aapne formula sahi likha aur steps saaf dikhaye — sirf final answer likhne se poore marks nahi milte. Har step likhein, chahe wo aasan hi kyun na lage. Formula likhna, values rakhna, calculation karna aur final answer — ye chaaron steps marks dilate hain.
Sabse aam galtiyan ye hain: x aur y ka kram ulta karna, section formula me m aur n ulta guna karna, kshetrafal me modulus bhoolna aur doori me vargmool lena bhoolna. Ek aur galti hai units na likhna — doori aur kshetrafal ke saath “units” aur “sq units” likhna zaroori hai. In chhoti baaton ka dhyan rakhne se 1-2 marks aaram se bach jaate hain.
Nirdeshank jyamiti class 10 notes ki taiyaari ke liye NCERT ke saare solved examples aur exercise ke sawaal zaroor karein. Maths ke doosre chapters ke notes ke liye [Prishthiya Kshetrafal aur Ayatan Chapter 12 notes](/prishthiya-kshetrafal-ayatan-class-10-notes/) aur [Tribhuj Chapter 6 notes](/tribhuj-class-10-notes-chapter-6/) bhi isi site par maujood hain. Exercise 7.1 distance par, 7.2 section formula par, 7.3 area par aur 7.4 miscellaneous sawaalon par hai. Jo student saari exercises kar leta hai, use exam me koi naya pattern nahi milta — bas wahi formula, naye numbers ke saath.
Aur bhi padhein: Prishthiya Kshetrafal Aur Ayatan Class 10 Notes aur Tribhuj Class 10 Notes Chapter 6.
Official Links
- NCERT Official Website: https://ncert.nic.in
- NCERT Class 10 Maths Book (Chapter 7): https://ncert.nic.in/textbook.php
- CBSE Official Website: https://www.cbse.gov.in
Aksar Puche Jaane Wale Sawaal (FAQs)
Nirdeshank jyamiti class 10 notes me sabse zaroori formula kaun se hain?
Nirdeshank jyamiti class 10 notes me 4 formula sabse zaroori hain: distance formula √[(x2 − x1)² + (y2 − y1)²], section formula ((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n)), midpoint formula ((x1 + x2)/2, (y1 + y2)/2) aur tribhuj ka kshetrafal ½|x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)|. In chaaron ko samajh kar likhne se poora chapter cover ho jaata hai.
Section formula me m aur n ulta kyun guna hote hain?
Kyunki point jis hisse ke kareeb hota hai, us taraf ke point ka bhaar zyada hota hai. Agar point A se 1:2 ke anupaat me hai to wo A ke kareeb hai, isliye B ka coordinate 1 se aur A ka coordinate 2 se guna hota hai. Ye “ulta” niyam hi section formula ki pehchaan hai, aur isi me sabse zyada galti hoti hai.
Collinearity kaise check karte hain?
Teen points se bane tribhuj ka kshetrafal nikaalein. Agar kshetrafal 0 aaye to points collinear hain, matlab ek hi rekha par hain. Agar kshetrafal 0 se zyada aaye to points tribhuj banate hain aur collinear nahi hain. Hamesha poora calculation dikhayein, sirf “area = 0” likhna kaafi nahi hai.
Distance formula aur Pythagoras pramey me kya rishta hai?
Distance formula Pythagoras pramey ka hi roop hai. Do points ke x-antaron aur y-antaron ko samkon tribhuj ki do bhujayein maan lein to unke beech ki doori karn (hypotenuse) hoti hai. Isliye doori = √(x-antar² + y-antar²). Ye samajh formula ko ratne se zyada mazboot banati hai.
Board exam me is chapter se kitne marks ke sawaal aate hain?
Aam taur par 3-4 marks ke sawaal aate hain, kabhi 5 marks ka bhi. Steps saaf likhne par poore marks milte hain — formula, values, calculation aur final answer, chaaron zaroori hain. Units (units/sq units) likhna mat bhoolein.
Aakhri Baat
Nirdeshank jyamiti class 10 notes ka poora chapter sirf 4 formula aur unki samajh par tika hai. Coordinate ka kram, section formula ka ulta guna aur kshetrafal ka modulus — in teen baaton ka dhyan rakhne se is chapter me poore marks aana mushkil nahi hai. NCERT ki exercises poori karein, practice questions khud hal karein aur exam me steps saaf likhein. Graph paper se darne ki zaroorat nahi hai — wo to bas numbers ka naksha hai.