Class 8 Maths Part 2 Ch 2 NCERT Solutions Hinglish mein

Ganita Prakash Grade 8 (pehla edition December 2025, 2026-27 mein chalne wala) ke Part 2 Chapter 2, The Baudhayana-Pythagoras Theorem, ke har sawaal ke jawab: text ke sawaal, har Figure it Out, Math Talk aur Try This, aur Find the Colours paheli. Page number book ke hisaab se hain (pages 33–54). Jo square roots poori sankhya nahi bante, unhe do decimal tak diya gaya hai. Short notes: Part 2 Chapter 2 ke notes.

Chapter 2 ka mind map: doguna aur aadha karna, sankhya √2, isosceles right triangles, theorem, Baudhayana triples, aur aage — Poora chapter ek page par

Square ko doguna aur aadha karna (pages 33–37)

Kya bhuja doguni karne se area doguna hoga? Nahi: naye square ka area 2 × 2 = 4 guna hoga.

Diagonal par bane square ka area doguna kyon? Konon se guzarti poorab-pashchim aur uttar-dakshin lines purane square ko 2 chhote right triangles mein aur diagonal wale square ko 4 mein kaatti hain, sab congruent, isliye uska area doguna hai.

Square ko doguna karna: diagonal par bana square 4 ek jaise triangles se bana hai, purana 2 se — Square ko doguna karna

Badhaayi gayi bhujaayein dotted square ke konon se kyon guzarti hain? Square ka diagonal uske kone ke angle ko aadha karta hai. Purane square ki bhujaayein diagonal se 45° par milti hain, isliye ve dotted square ke 90° konon ko aadha karti hain, aur square ke kone ka bisector saamne wale kone se guzarta hai.

Chhote triangles congruent kyon hain? Har ek ek jaise chhote square ka diagonal se kata aadha hai: sabki do barabar bhujaayein (diagonal ka aadha) right angle par milti hain, isliye SAS se congruent.

Kaagaz wali activity. Tukde 5, 6, 7 aur 8, yaani Square 2 ko dono diagonals se kaatkar bane chaar chauthaai, Square 1 ki chaaron bhujaon par lagaane se doguna area wala square banta hai.

Andar wala square bade ka aadha kyon? Beech ke binduon se guzarti lines bade square ko 8 congruent triangles mein kaatti hain; tircha square unmein se 4 rakhta hai.

Kya aadhi bhuja wale square ka area aadha hoga? Nahi, sirf chauthaai: aise 4 square asli ko bharte hain.

PQRS aadhe area ka square kyon hai? Chaaron kone wale triangles congruent hain (SAS: right angle par milti bhujaon ke aadhe), isliye PQ = QR = RS = SP. Har kone wala triangle isosceles aur right-angled hai, to uske baaki angles 45° hain, aur PQRS ka har angle 180° − 45° − 45° = 90°. PR aur QS jodne par PQRS chaar aise triangles mein bantta hai jo mude tukdon ke congruent hain, to tukde PQRS ko theek dhak lete hain: wo kaagaz ka aadha hai.

Modkar square ko aadha karna: kone PQRS par mudte hain, jiska area aadha hai — Modkar square ko aadha karna

Hypotenuse aur √2 (pages 37–41)

1-1 right triangle ka hypotenuse: √2, kyunki us par bane square ka area 2 hai. Bounds: 1 < √2 < 2; 1.4 < √2 < 1.5; 1.41 < √2 < 1.42; 1.414 < √2 < 1.415. Terminating decimal? Nahi: uska varg non-zero ank par khatm hoga. Fraction? Nahi: 2n² = m² asambhav hai.

Figure it Out (pages 39–40)

1. Har square ko ek diagonal se kaatkar chaar right triangles banao. Unke right angles ek beech ke bindu par milaao aur lambi bhujaayein baahar rakho: ek square banta hai jiski bhuja diagonal hai, aur area kisi bhi square ka doguna.

Figure it Out 1: do squares ke chaar triangles se doguna area wala ek square — Figure it Out 1: chaar triangles

2. c² = 2a². (i) a = 3: c = √18 ≈ 4.24, to 4.2 < c < 4.3 (4.2² = 17.64, 4.3² = 18.49). (ii) a = 4: c = √32 ≈ 5.66, to 5.6 < c < 5.7. (iii) a = 6: c = √72 ≈ 8.49, to 8.4 < c < 8.5. (iv) a = 8: c = √128 ≈ 11.31, to 11.3 < c < 11.4. (v) a = 9: c = √162 ≈ 12.73, to 12.7 < c < 12.8.

3. Hypotenuse par bane square ka area 10² = 100 hai, jo bhuja par bane square ka doguna hai, isliye a² = 50 aur har bhuja √50 ≈ 7.07 (7.0 aur 7.1 ke beech).

Do squares jodna (pages 41–47)

Kya do barabar squares ke liye tareeka mel khaata hai? Haan: barabar chhoti bhujaon par hypotenuse square ka diagonal hai, jo theek pehle wala doguna karne ka niyam hai.

Chaar bhujaon wali aakriti square kyon hai? Uski chaaron bhujaayein congruent triangles ke hypotenuse hain, isliye barabar hain. Har kone par do triangles ke angles x aur 90° − x ek seedhi line (180°) par hain, to bachta hai 180° − 90° = 90°. Chaar barabar bhujaayein aur chaar right angles matlab square.

Do squares ek ban jaate hain: do kaat aur do khiske triangles a aur b bhujaon ke squares ko c bhuja ka square bana dete hain — Do squares ek ban jaate hain

Figure it Out (page 47)

1. 5² + 12² = 169, to hypotenuse 13 cm.

2. 17² − 8² = 289 − 64 = 225, to teesri bhuja 15 cm.

3. Teen guna: aisa right triangle banao jiski chhoti bhujaayein square ki bhuja a aur uska diagonal a√2 hon; hypotenuse par bane square ka area a² + 2a² = 3a². Paanch guna: chhoti bhujaayein a aur 2a lo; hypotenuse par square ka area a² + 4a² = 5a².

Teen guna aur paanch guna square: a aur a√2, aur a aur 2a bhujaon wale right triangles — Teen guna aur paanch guna square

4. (i) c = √74 ≈ 8.60. (ii) c = √208 ≈ 14.42. (iii) b = √(225 − 81) = √144 = 12. (iv) c = √193 ≈ 13.89. (v) c = √(2.25 + 12.25) = √14.5 ≈ 3.81.

Baudhayana triples (pages 48–50)

20 tak ke triples: (3, 4, 5), (6, 8, 10), (5, 12, 13), (9, 12, 15), (8, 15, 17) aur (12, 16, 20). (30, 40, 50) aur (300, 400, 500) triples hain: 900 + 1600 = 2500 aur 90,000 + 1,60,000 = 2,50,000. Pattern: (6, 8, 10), (9, 12, 15) aur (12, 16, 20) = (3, 4, 5) ka 2, 3 aur 4 guna. (5, 12, 13) primitive hai; 20 tak ke baaki primitive triples (3, 4, 5) aur (8, 15, 17) hain. Unke scaled roop neeche hain; koi primitive nahi, kyunki k teeno sankhyaon ko kaatta hai.

Primitive triples (3, 4, 5), (5, 12, 13) aur (8, 15, 17), har ek ke paanch scaled roop — Primitive triples aur unke guna

Kya (a/f, b/f, c/f) triple hai? Haan: a² + b² = c² ko f² se bhaag dene par (a/f)² + (b/f)² = (c/f)². (9, 12, 15) aur f = 3 se (3, 4, 5), aur 9 + 16 = 25. n wi odd sankhya 2n − 1 hai, aur pehli n − 1 odd sankhyaon ka jod (n − 1)² hai.

Figure it Out (page 50)

1. Odd varg 49, 81, 121, 169 aur 225 se: (7, 24, 25), (9, 40, 41), (11, 60, 61), (13, 84, 85) aur (15, 112, 113).

Odd vargon se triples: 9, 25, 49 aur aage se (3, 4, 5), (5, 12, 13), (7, 24, 25) waghera — Odd vargon se triples

2. Nahi. Ek chhoti bhuja, n − 1, hypotenuse n se ek kam hai, aur lagaataar sankhyaon ka 1 se bada koi common factor nahi hota, isliye is tareeke ka har triple primitive hai.

3. Haan, jaise (8, 15, 17), (20, 21, 29), (12, 35, 37) aur (28, 45, 53): inmein koi bhuja hypotenuse se ek kam nahi.

Upyog (pages 52–54)

Lilavati ka kamal: 3² + x² = (x + 1)² se x = 4, yaani jheel 4 unit gehri hai.

Bhaskaracharya ka kamal: x aur 3 bhujaon aur x + 1 hypotenuse wala right triangle batata hai ki jheel 4 unit gehri hai — Bhaskaracharya ka kamal

Figure it Out (pages 52–54)

1. 5√2 = √50 ≈ 7.07 cm.

2. Right angle ke aas-paas 7 aur 9: hypotenuse √130 ≈ 11.40. 4 aur 10: hypotenuse √116 ≈ 10.77. Bhuja 40, hypotenuse 41: teesri bhuja √(1681 − 1600) = 9. Bhuja 10, hypotenuse √200: teesri bhuja √(200 − 100) = 10. 10 aur √150: hypotenuse √250 ≈ 15.81. Bhuja 27, hypotenuse 45: teesri bhuja √(2025 − 729) = 36.

3. Diagonals ek doosre ko right angle par aadha karte hain, to 12 aur 35 bhujaon wale right triangles bante hain: bhuja = √(144 + 1225) = √1369 = 37 unit.

Question 3: rhombus ke aadhe diagonals 12 aur 35 se bhuja 37 — Question 3: rhombus

4. Haan. c² = a² + b², jo a² aur b² dono se bada hai, isliye c, a aur b dono se bada hai.

5. Sahi: kisi bhi non-primitive triple ko uski sankhyaon ke greatest common factor se bhaag do to primitive triple milta hai, jiska wo scaled roop hai.

6. Jaise 3 × 4 (diagonal 5), 6 × 8 (10), 5 × 12 (13), 8 × 15 (17) aur 7 × 24 (25).

7. Chahiye area 7² − 5² = 49 − 25 = 24. Aisa right triangle banao jiska hypotenuse 7 aur ek bhuja 5 ho: 5 unit ki line, uske ek sire par perpendicular, aur doosre sire se 7 radius ka arc jo perpendicular ko kaate. Perpendicular bhuja √24 hai, aur us par bane square ka area 24.

8. (i) (a) Haan: tircha square jiski bhuja 1 aage aur 1 upar ke dots jodti hai (bhuja² = 1 + 1 = 2). (b) Nahi: 3 do vargon ka jod nahi. (c) Haan: 2 × 2 square. (d) Haan: tircha square jiski bhujaayein 1 aage aur 2 upar jaati hain (bhuja² = 1 + 4 = 5). (ii) Theek wahi sankhyaayein jo do vargon ka jod p² + q² hain: 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25 waghera.

Question 8: dot grid par 1, 2, 4 aur 5 area wale squares — Question 8: dot grid par squares

9. Altitude triangle ko do congruent right triangles mein baantti hai (RHS: hypotenuse 6 ki barabar bhujaayein aur altitude dono mein saajha), isliye base ko 3 aur 3 mein aadha karti hai. Phir h² = 6² − 3² = 27, h = 3√3 ≈ 5.20, aur area ½ × 6 × 3√3 = 9√3 ≈ 15.59 square unit.

Question 9: 6 bhuja wale equilateral triangle ki altitude √27 aur area 9√3 — Question 9: equilateral triangle

Find the Colours (page 54)

RED label wala dabba kholo. Uska label galat hai, to usmein neeli ya hari gendein hain. Agar neeli hain, to BLUE label wale mein neeli nahi ho sakti (label galat hai, aur neeli pehle hi mil gayi), to usmein hari hain, aur GREEN wale mein laal. Agar hari hain, to GREEN wale mein hari nahi ho sakti, to usmein neeli hain, aur BLUE wale mein laal. Ek dabba kaafi hai.

Short notes aur zyada practice topic pages par hai: theorem kya hai?, Pythagorean triples kya hain? aur root 2 kya hai?.

NCERT Solutions Class 8 Maths Part 2 Chapter 2: The Baudhayana-Pythagoras Theorem (Hinglish)