NCERT Solutions Class 8 Maths Part 2 Chapter 4: Exploring Some Geometric Themes (Hinglish)

Ganita Prakash, Grade 8 Part 2, ke Chapter 4, Exploring Some Geometric Themes, ke har sawaal ke jawaab: text ke sawaal, Try This aur Math Talk prompts aur saare Figure it Out. Page numbers book ke hisaab se hain (pages 70–102). Chhote notes: Part 2 Chapter 4 ke notes.

Fractal (pages 70–75)

Sierpinski carpet: pattern. Bache varg 1, 8, 64, 512, … (Rₙ = 8ⁿ), aur chhed 0, 1, 9, 73, … (Hₙ₊₁ = Hₙ + Rₙ, isliye Hₙ = 1 + 8 + … + 8ⁿ⁻¹ = (8ⁿ − 1) ÷ 7).

Dikhao ki madhya-bindu jodne se 4 ek jaise samabahu tribhuj bante hain. Tribhuj ABC mein madhya-bindu D (AB par), E (BC par) aur F (CA par) hon, to AD = AF, dono barabar bhujaaon ke aadhe, aur ∠A = 60°. Barabar bhujaaon ke beech 60° wale samdvibahu tribhuj ke baaki do kon bhi 60° hote hain, isliye ADF samabahu hai, bhuja AB ki aadhi. B aur C par bhi yahi. Beech ke tribhuj DEF ki bhujaayein DF, DE aur EF hain, har ek kone ke kisi tribhuj ki bhuja, isliye ye bhi usi bhuja wala samabahu hai. Chaaron ek jaise hain.

Figure it Out (page 72), 1. Neeche chitra dekho: kadam 0 poora tribhuj, kadam 1 ek chhed ke chaaron or teen tribhuj, kadam 2 nau tribhuj. 2. Bache tribhuj: 1, 3, 9, 27, …, 3ⁿ. Chhed: 0, 1, 4, 13, …, 1 + 3 + … + 3ⁿ⁻¹ = (3ⁿ − 1) ÷ 2. 3. Carpet: har kadam 8/9 kshetrafal rakhta hai, isliye kadam n par (8/9)ⁿ varg ikai. Triangle: har kadam 3/4, isliye (3/4)ⁿ varg ikai.

Koch snowflake, Figure it Out (page 73), 1. Chitra dekho. 2. Har bhuja 4 bhujaayein banti hai, isliye bhujaayein 3, 12, 48, …, 3 × 4ⁿ. 3. Har nayi bhuja 1/3 lambi hai, isliye kadam n par bhuja (1/3)ⁿ aur parimaap 3 × 4ⁿ × (1/3)ⁿ = 3 × (4/3)ⁿ ikai.

Man mein banaao (pages 75–77)

1. Man mein karo: jaise RAVI ulta I, V, A, R. 2. 45° ghooma ek chhota varg, aadhe kshetrafal ka. Chaaron kone ke tukde samkon samdvibahu tribhuj hain; unke samkon wale kone ek bindu par milaao to bache varg jitna hi ek varg banta hai. 3. Niyamit shatbhuj. 4. Barabar konon (135°) wala ashtabhuj, par uske kate kinaare bache beech ke tihaai hisson se lambe hain.

5–12. Table dekho: saamne se cube (varg); gola, ya upar se cylinder ya cone (vritt); bagal se cone ya pyramid (tribhuj); cylinder (aayat aur vritt); cone (vritt aur tribhuj); tribhujiya prism (aayat aur tribhuj); baalti jaisa aakaar, nok kata cone (samlamb chaturbhuj aur vritt); panchbhujiya prism (panchbhuj aur aayat). Kya ek hi thos? Nahi: zyaadatar sharton par kai thos fit hote hain, jaise square pyramid bhi bagal se tribhuj dikhta hai.

Prism aur pyramid (page 79)

10-bhuja prism: 12 falak, 30 kinaare, 20 sheersh; n-bhuja: n + 2 falak, 3n kinaare, 2n sheersh. 10-bhuja pyramid: 11 falak, 20 kinaare, 11 sheersh; n-bhuja: n + 1 falak, 2n kinaare, n + 1 sheersh.

Net (pages 79–84)

Figure it Out (page 80), 1. Cube ke net: (ii), (iii), (iv) aur (vi). (i) aur (v) nahi: modne par do varg ek hi falak par aate hain aur ek falak khula reh jaata hai.

2. Saare 11 net neeche bane hain. 3. (i) Chaar aayaton ki patti, 5 × 3, 5 × 1, 5 × 3 aur 5 × 1 (cm mein), jinmein se ek ke dono siron par 3 × 1 ka aayat. (ii) Yahi 6 × 3, 6 × 2, 6 × 3 aur 6 × 2 ke aayaton ke saath, aur siron par do 3 × 2 ke aayat.

Niyamit tetrahedron ke net. Pehla chitra (chaar ka bada tribhuj) aur teesra (chaar ki seedhi patti) net hain. Doosra modne par do tribhuj ek falak par aate hain, aur chauthe mein paanch tribhuj hain. Aur koi nahi. Maap wale net: tetrahedron ke liye 6 cm bhuja ke chaar samabahu tribhuj, kisi bhi tarah; square pyramid ke liye 6 cm ka varg, har taraf 6 cm aadhaar wala samabahu (ya samdvibahu) tribhuj.

Cylinder ka net. Aayat ki lambaai 2πr, yaani aadhaar ki paridhi, aur chaudaai h, yaani oonchaai. Cone ka net: O ko kendra maanne wale l (tirchhi oonchaai) trijya ke vritt ka sector, aur aadhaar ke liye ek vritt. O ko chaap ke kendra se hatakar rakhne par tirchhi lambaaiyaan alag hoti hain, aur net ek tircha (oblique) cone banata hai jiski nok aadhaar ke kendra ke upar nahi. Tribhujiya prism: teen aayat saath-saath (4 cm bhuja ke samabahu sire aur 8 cm lambaai ke liye teen 8 × 4 ke aayat) aur har sire par ek samabahu tribhuj.

Octahedron: book ke net se mude aath samabahu tribhuj aadhaar se jude do square pyramid banaate hain. Gole ka net? Sambhav nahi: gend par lapeta koi bhi samtal tukda silvat, khaali jagah ya overlap chhodta hai, kyunki gola har disha mein muda hai.

Cuboid par sabse chhota raasta (pages 84–87)

Pehla udaaharan: diya raasta net par seedhi rekha banta hai, isliye sabse chhota hai. Doosra udaaharan: diye raaste net par mudte hain, isliye sabse chhote nahi; net par seedhi rekha hai. Pakka kaise? Net par sabse chhota raasta seedhi rekha hai, aur dabbe ka har raasta net par usi lambaai ka hai. Teesra udaaharan (8 cm, 4 cm, 4 cm ka dabba): pehli tarah kholne par seedhi rekha net se baahar jaati hai, isliye asli raasta nahi; doosri tarah kholne par rekha net par rehti hai aur sabse chhota raasta milta hai. Mushkil udaaharan (30 × 12 × 12 ka dabba): alag tareeke alag lambaai dete hain, ek ka 42 cm aur doosre ka √(24² + 32²) = 40 cm; sabse chhota raasta 40 cm hai. Kya poora ho gaya? Tabhi, jab kholne ke saare tareeke milaa liye jaayein.

Prakshep (pages 87–92)

p aur l ki tulna. AECD aayat hai kyunki AD aur EC dono tal par lamb hain aur AE ko DC ke samaantar kheencha gaya, isliye AE = DC = p. Tribhuj AEB mein ∠AEB = 90°, isliye AB = l karn hai: p ≤ l. Barabar tab jab rekha tal ke samaantar ho.

Varg ke prakshep: varg (tal ke samaantar), aayat (ek bhuja ke charon or jhuka), samaantar chaturbhuj (aur tarah jhuka) ya rekhakhand (kinaare se). Samaantar chaturbhuj ka: hamesha samaantar chaturbhuj ya rekhakhand, kabhi aur tarah ka chaturbhuj nahi, kyunki samaantar rekhaayein samaantar rekhaaon mein prakshipt hoti hain. Niyamit n-bhuj ka: n bhujaaon wala bahubhuj (niyamit sirf tal ke samaantar hone par), samaantar bhujaayein samaantar, ya rekhakhand. Cube aur cone: cube varg, aayat ya shatbhuj deta hai; cone vritt (upar se), tribhuj (bagal se) ya beech ka gol aakaar. Usi prakshep wali doosri vastu: zyada jhuki lambi rekha, sire se dekha ooncha cuboid, aur cone ke liye usi oonchaai aur aadhaar-chaudaai ka square pyramid (bagal se wahi tribhuj) ya cylinder (upar se wahi vritt).

Figure it Out (page 92), 1. Teeno rekhaaon ka saamne ka drishya ek jaisa hai, yaani ve barabar chaudaai gherti hain. Bagal ka drishya dikhaata hai ki har rekha peechhe kitni jaati hai, aur upar ka drishya tirchhi lambaai hai: (upar ka drishya)² = (saamne ka)² + (bagal ka)². Aam taur par teeno drishyon ki lambaaiyon ke vargon ka jod rekha ki asli lambaai ke varg ka dugna hota hai.

2. Falak talon ke samaantar rakhkar: cube: teen barabar varg. Cuboid: teen aayat. Parallelepiped: jhukaav ke hisaab se samaantar chaturbhuj ya aayat. Khada cylinder: aayat (saamne), vritt (upar), aayat (bagal). Khada cone: tribhuj, kendra wala vritt, tribhuj. Aayataakaar falak par leta tribhujiya prism: tribhuj (saamne), aayat (upar), aayat (bagal). Square pyramid: tribhuj, vikarnon wala varg, tribhuj.

3. Naapne ka jug: aakhri pankti (jug, tonti aur handle wala vritt, jug). Keep: chhathi pankti. Hathauda: saatvi pankti. Car: pehli pankti. Fisalpatti: teesri pankti (patla aayat, lambi patti, dhalwaan bagal). Kursi: chauthi pankti. Pankha: paanchvi pankti. Tiffin box: doosri pankti (dhakkan wala dabba, handle wala vritt, handle wala dabba).

Parchhaaiyaan (page 94)

Parchhaai prakshep jaisi par badi hoti hai. Torch ko vastu ke paas laane par parchhaai badi hoti hai, kyunki torch se kirnein failti hain aur deewar tak pahunchne se pehle zyada fail jaati hain; door le jaane par kirnein lagbhag samaantar ho jaati hain aur parchhaai prakshep jaisi ho jaati hai. Dhoop mein samaantar chaturbhuj ki parchhaai hamesha samaantar chaturbhuj hoti hai.

Figure it Out (pages 95–97)

1. Har teer ki disha mein dekho. Pehli aakriti (samtal L): saamne lamba aayat, upar L, bagal chhota aayat. Doosri (oonche khambhe ke paas neecha block): saamne ooncha aayat aur uske paas chhota varg, upar vargon ka L, bagal ooncha aayat aur peechhe neecha kadam. Teesri (samtal zigzag): saamne lamba aayat, upar Z jaisa kadam, bagal aayat. Chauthi (peechhe seedhidaar deewar wala slab): saamne chauda aadhaar aur baayein ooncha hissa, upar kadamon par rekhaaon wala aayat, bagal L. Paanchvi (khada T): saamne daayein ubhre varg wala khamba, upar chhoti pankti, bagal ek khamba. Chhathi (aadhaar par khaanchedaar deewar): saamne aadhaar par do khambe aur beech mein khaali, upar rekhaaon wala aayat, bagal L.

2. (i) Bagal se 4 cube ka ek khada khamba; upar se 3 ki pankti. (ii) C ko peechhe khada karo aur uski upari parat par aage ki or 7 cube jodo, taaki upar se upari parat A padhi jaaye: 15 cube. (iii) Baayein khambe se sate doosre star par 2 cube aur jodo: baayein se ab F dikhta hai, kul 17 cube. (iv) Doosre teen akshar bhi aise ban sakte hain, bas har drishya ki chaudaai aur oonchaai mel khaayein.

3. Thos (ii): peechhe 3 aur 2 ke oonche dher, beech mein ekal cube ki pankti aur aage baayein ek cube. 4. (i)–(iii): 9 cube; (iv)–(vi): 7 cube; (vii)–(ix): 8 cube, neeche ke oonchaai-grid ke hisaab se.

5. 10 cube dikhte hain. Agar har cube neeche ke cubon par tika hai, to dher 10, 6, 3 aur 1 ki parton wala tribhujiya pyramid hai: 20 cube.

6. Varg, aayat ya shatbhuj (kone par santulit hone par niyamit shatbhuj).

Isometric prakshep (pages 97–102)

Prakshipt kinaare barabar kyon? Kone par santulit cube ka lamba vikarn khada hota hai, aur us par ek-tihaai chakkar ghumaane par cube waisa hi dikhta hai. Isliye upar ke kone ke teeno kinaare, aur aise hi saare 12 kinaare, utne hi jhuke hain aur ek hi lambaai mein prakshipt hote hain. 2 × 2 × 2 ka cube har aksh par do ikai chalkar banta hai, jaisa dikhaaya hai.

Isometric kaagaz itna kaam ka kyon? Samaantar kinaare samaantar rehte hain aur lambaai, gehraai aur oonchaai par barabar kadam barabar rehte hain, isliye chitra aakaar ke anupaat bachaata hai.

Figure it Out (page 100), 1. Haan, paanch samtal aakaaron ke alaawa teen aur hain: ek kona jismein teen cube ek beech wale cube se teen dishaon mein jude hon, aur do mude aakaar jo ek-doosre ke darpan-pratibimb hain. 2. Har kinaara oonchaai (|), lambaai ya gehraai ki rekhaaon par ikaiyaan ginte hue banao. 3. Haan: gend ek hi disha mein ghoomti lagti hai phir bhi shuruaat par laut aati hai. Chitra ka har hissa sambhav hai, par poora chakra sambhav nahi; isometric chitra gehraai chhupaata hai, isliye door ke bindu jude lag sakte hain. 4. (i) Nahi; cube ka asli model sirf ek theek drishti-bindu se aisa dikh sakta hai. Iske saamne, upar aur bagal ke profile har ek uski teen pattiyon mein se do se bana L hote. (ii) Har patti ek isometric disha mein banao. (iii) Bhram isliye kaam karta hai ki isometric chitra mein gehraai ki jaankaari nahi hoti: hum har kona sahi padhte hain, aur dimaag maan leta hai ki chitra mein milte sire jagah mein bhi milte hain.