Class 6 Maths Chapter 9 NCERT Solutions Hinglish mein: Symmetry

Ganita Prakash (Reprint 2026-27) ke Chapter 9, Symmetry, ke har “Figure it Out” sawal ke jawab, saath mein in-text sawal, tiles wali activity aur game. Page numbers book se milte hain. Jahan sawal mein banana hai, wahan jawab batata hai ki kya banana hai aur poori figure kaisi dikhti hai. Short notes: Chapter 9 notes.

Hamare aas-paas symmetry (pages 217–218)

• Titli: uske baayein aur daayein pankh ek doosre ke mirror image hain.
• Rangoli: laal pankhudiyan centre ke chaaron taraf chauthai chakkar (90°) ghumane pe apni jagah wapas aati hain, aur baaki hisse bhi.
• Pinwheel: beech ke hexagon se shuru karo. Uski har side pe ek hi shape chipki hai, do equilateral triangles se bana rhombus, aur har rhombus ek hi taraf jhuka hai. Hexagon ke chaaron taraf chalne pe har rhombus pichhle wale ko 60° ghumane se banta hai.
• Baadal: koi hissa tay pattern mein nahi dohraata, isliye tasveer symmetrical nahi.
• Taj Mahal aur gopuram: dono ke beech mein khadi line of symmetry hai; baayan aadha daayein aadhe se mel khaata hai.

Figure it Out: lines of symmetry (page 219)

1. Haan. Phool mein 6 lines of symmetry hain, rangoli mein 4 aur titli mein 1. Pinwheel mein koi line of symmetry nahi (usme rotational symmetry hai), aur baadal mein bhi koi nahi.
2. • Pehli figure (teer jaisi shape): 1 line, uske nukeele baayein kone se daayein taraf ke khaanche tak ki tirchhi line.
   • Kite (patang): 1 line, uske upar aur neeche ke kono se.
   • Tirchhe upar wali paanch sides ki figure: koi line of symmetry nahi.
   • L-shape: 1 line, uske bahari kone aur andar ke kone se guzarti tirchhi line (dono bhujaayein ek jaisi hain).
   • Triangle: koi line of symmetry nahi; uski saari sides alag hain.

Square aur rectangle modna (page 221)

Kya square ko modne ka koi aur tareeka hai jisse hisse mel khaayein? Nahi. Khada mod, leta mod aur dono diagonal mod hi hain, isliye square mein 4 lines of symmetry hain.

Kai lines of symmetry wali teen figures: snowflake jaise star mein 6, chhe pankhudiyon wale phool mein 6, aur loop wali square gaanth mein 4 (khadi, leti aur dono diagonals).

Jo rectangle square nahi, kya uska diagonal line of symmetry hai? Nahi. Rectangle ko diagonal pe modo to dono hisse mel nahi khaate: kone bahar nikle rehte hain.

Reflection (page 222)

Square ABCD mein (A upar-baayein, B upar-daayein, C neeche-daayein, D neeche-baayein):

• Diagonal AC mein reflect karna: A aur C wahin rehte hain, kyunki wo line pe hain; B, D ki jagah aur D, B ki jagah jaata hai.
• Leti line of symmetry mein reflect karna: A aur D jagah badalte hain, aur B aur C jagah badalte hain.

Syahi ke dhabbe (ink blot devils): pattern symmetric hota hai, aur uski line of symmetry mod hai. Aam taur pe koi aur line nahi hoti jis pe modne se do ek jaise hisse banein.

Kaagaz modna aur kaatna (page 223)

Khula hua kaagaz: mod se laga trapezium jaisa cut-out kholne pe chhe sides ka chhed banta hai, jo mod pe jude do trapeziums se bana hai. Iski line of symmetry mod ki line hai.

Sajaawati patti: har star aur har diamond ke beech se khadi line of symmetry hai, padosi stars ke theek beech khadi lines hain, aur patti ke beech ki leti line bhi line of symmetry hai.

Figure it Out: punching game aur kaagaz kaatna (pages 223–230)

1. Kaagaz kahan moda gaya tha? a. Beech ki khadi line pe. b. Neeche-baayein kone se upar-daayein kone tak ke diagonal pe. c. Beech ki leti line pe. d. Kaagaz do baar moda gaya, pehle khada aur phir leta (ya ulta), isliye ek chhed se chaar chhed bane.

2. Baaki chhed: har gayab chhed diye gaye chhed ka mirror image hai: line ke doosri taraf, usse barabar doori pe, line ke saath 90° wali line pe.

   • a. Upar-baayein kone ke theek neeche wala chhed us kone ke theek daayein, upar ki side ke saath jaata hai.
   • b. Neeche-daayein wala chhed upar-daayein, line se utna hi upar jaata hai.
   • c. Upar ke paas line ke theek baayein wala chhed line ke theek daayein, utni hi oonchai pe jaata hai.
   • d. Kinaare ke paas tirchhi line ke theek upar wala chhed line ke theek neeche, kinaare pe milti jagah pe jaata hai.
   • e. Upar ke paas tirchhi line ki ek taraf wala chhed doosri taraf ki milti jagah pe jaata hai.

3. Ye hissa sirf dikhata hai ki khada mod aur leta mod kaise banaya jaata hai.

4. Chhed ka andaaza:

   • a. Bow (fiite) jaisa chhed: mod pe patla, dono taraf lehardar sire.
   • b. Riban banner jaisa chhed, dono siron pe V-khaanche ke saath.
   • c. Beech ki khadi line pe do rectangle chhed, ek upar ke aadhe mein aur ek neeche ke aadhe mein, aur chaar khaanche: do upar ki side mein aur do neeche ki side mein.
   • d. Beech mein mod pe ek square chhed, aur har bagal wali side mein usi oonchai pe ek khaancha.

5. Ek seedhi katai:

   • a. Square ko aadha modo, phir dobara aadha, taaki uska centre band kona ban jaaye; phir us kone se guzarte diagonal pe ek baar aur modo. Mude kaagaz pe kisi side ke parallel ek seedhi katai kholne pe aisa square chhed deti hai jiski sides kinaaron ke parallel hon.
   • b. Square ko aadha aur phir dobara aadha modo, phir band kone pe dono modon se barabar tirchhi seedhi katai karo. Kholne pe kone pe khada square chhed banta hai.

   Dono mein chhed square hai: uski chaaron sides ek hi katai ki copy hain, isliye barabar hain, aur uske angles 90° hain.

6. a. Tirchhe square mein 4 lines of symmetry hain aur aath kono wale star mein 8. b. Equilateral triangle mein 3. c. Regular hexagon mein 6.

7. Rhombus wali figures: pehli (do pe ek rhombus) mein 1 line, khadi; teen ki row, chaar ka diamond aur chhe ka group, sab mein 2, khadi aur leti. Squared paper wali figures: square ke andar square mein 4; aath sides wali figure mein 2, khadi aur leti; paanch sides wali figure mein 1, uske nukeele sire se guzarti leti line; chaar nok wale star mein 4.

8. Kolam mein 6 lines of symmetry hain: leti line aur aamne-saamne ke star patterns se guzarti do tirchhi lines, aur khadi line aur unke beech ki do tirchhi lines.

9. a. Sirf do barabar sides wala isosceles triangle, jaise 5 cm, 5 cm aur 3 cm sides: 1 line. b. Equilateral triangle: 3 lines. c. Scalene triangle, saari sides alag, jaise 3 cm, 4 cm aur 6 cm: koi line nahi. Do lines possible nahi. Triangle ki har line of symmetry uski do sides barabar karti hai. Do lines teeno sides barabar kar dengi, aur tab triangle mein teen lines hongi.

10. Curved kinaare ke saath: a. Semicircle, ya dil ki shape: theek 1 line. b. Do barabar arcs se bani aankh jaisi shape, ya running track jaisi shape: theek 2 lines. c. Har side pe semicircle wala square, chaar pankhudiyon wale phool jaisa: theek 4 lines.

11. Har kona neeli line ke paar copy karo, upar wale diagram ki tarah squares gin ke. (c) aur (f) ke liye book ko itna ghumao ki neeli line khadi dikhe.

    • b. Poori figure line ke upar aur neeche ek jaisi hai: upar aur neeche ek tower, baayein ek step aur daayein ek nok.
    • c. Diagonal ke upar ki seedhiyan uske neeche copy hoti hain, aur figure seedhi jaisi shape mein band ho jaati hai jo diagonal pe modne se apne aap pe aati hai.
    • d. Chhota ghar: chhote rectangle pe triangle ki chhat.
    • e. Paanch sides wali shape aur uski ulti copy, line pe judi hui.
    • f. Shape aur diagonal ke doosri taraf uska mirror image, line ke paas ek doosre pe chadhe hue.

12. Drawing ko ek neeli line mein reflect karo, phir sab kuch doosri line mein.

    • a. Ek square.
    • b. Ek square jiski har side ke beech se ek chhoti triangle jaisi nok bahar nikli ho.
    • c. Ek lamba seedhiyon wala figure, jiske baayein-daayein aur upar-neeche ek jaise khaanche hon.
    • d. Aath sides wali figure: ek rectangle jiske chaaron kone kate hon.
    • e. Cross jaisi figure, upar aur neeche nukeele khaanche ke saath.
    • f. Centre pe milte chaar ek jaise tukde, dono dishaon mein bow tie jaise.

13. Har grid mein dots ko column (baayein se 1 se 6) aur row (upar se 1 se 6) se gino, yaani (3, 1) matlab column 3, row 1. Har figure ka ek jawab:

    • Pehli: diya hai (1, 2)–(5, 1) aur (1, 2)–(5, 6). Row 3 aur 4 ke theek beech leti line of symmetry ke saath (5, 1)–(1, 5) aur (1, 5)–(5, 6) kheencho.
    • Doosri: diya hai (3, 1)–(1, 4), (1, 4)–(2, 6) aur (2, 6)–(4, 6). Column 3 se guzarti khadi line ke saath (3, 1)–(5, 4) aur (5, 4)–(4, 6) kheencho; pentagon banta hai.
    • Teesri: diya hai (1, 1)–(3, 2), (3, 2)–(5, 1), (5, 1)–(5, 3) aur (5, 3)–(3, 6). Column 3 se guzarti khadi line ke saath (1, 1)–(1, 3) aur (1, 3)–(3, 6) kheencho.
    • Chauthi: diya hai (1, 1)–(5, 1), (5, 1)–(5, 2) aur (5, 2)–(2, 5). (1, 1) aur (5, 5) se guzarti tirchhi line ke saath (1, 1)–(1, 5) aur (1, 5)–(2, 5) kheencho.
    • Paanchvi: diya hai (1, 2)–(3, 4) aur (1, 2)–(3, 6). Column 3 se guzarti khadi line ke saath (3, 4)–(5, 2) aur (5, 2)–(3, 6) kheencho.
    • Chhathi: diya hai (1, 1)–(3, 1), (3, 1)–(6, 3) aur (6, 3)–(3, 5). Row 3 se guzarti leti line ke saath (1, 1)–(1, 5) aur (1, 5)–(3, 5) kheencho.

Rotational symmetry: in-text sawal (pages 230–235)

• Kya windmill 90° se kam ghumane pe waisi dikhti hai? Nahi. Uske angles of symmetry 90°, 180°, 270° aur 360° hain.
• Theek chaar angles of symmetry wali koi aur shape: square, plus ka nishaan, ya chaar blades wala pankha.
• Square: iske 4 angles of symmetry hain, 90°, 180°, 270° aur 360°, aur apne aap pe aane ke liye 90° ghumana padta hai. Centre of rotation wo point hai jahan diagonals milte hain.
• Patti (example): aadhe chakkar ke baad iski lambi side upar hoti hai, isliye ye original pe nahi aati; sirf 360° ke baad mel khaati hai. Isme rotational symmetry nahi.
• 4 radial arms wali figure: 4 angles of symmetry, 90°, 180°, 270° aur 360°. Arms ke beech ke angles badlo to 4 angles of symmetry nahi rehte, kyunki chauthai chakkar mein har arm agli pe aani chahiye, isliye chaaron angles 90° hone chahiye.
• Sirf do angles of symmetry: har arm ka sira ek hi taraf modo, jaisa book ki figure mein hai, ya do aamne-saamne ki arms ko baaki do se lamba karo. Tab sirf 180° aur 360° chalte hain.
• Teen radial arms: alag-alag angles ke saath sirf 360° chalta hai. Teeno angles 360° ÷ 3 = 120° hon to angles of symmetry 120°, 240° aur 360° hain.
• Paanch aur chhe radial arms: padosiyon ke beech 72° wali 5 arms se 72°, 144°, 216°, 288° aur 360° milte hain; 60° wali 6 arms se 60°, 120°, 180°, 240°, 300° aur 360°.
• Saat radial arms: sabse chhota angle of symmetry 360° ÷ 7 = 51 3/7° hai, jo poora number nahi.

Figure it Out: angles of symmetry (pages 235–236)

1. a. Plus jaisi figure: 90°, 180°, 270°, 360°. b. Dono siron pe ek hi taraf chhote squares wali line: sirf 360° (aadhe chakkar ke baad squares doosri taraf honge). c. Beech mein judi do lambi lines wali figure: 180°, 360°.
2. Ek se zyada angle of symmetry: 90° pe do diameters wala circle, 120° ki doori pe teen radii wala circle, chaar mude blades, 90° pe kaatti do lines ka cross, aur paanch nok wala star. Dot wala triangle (uski saari sides barabar nahi) aur do semicircles wali figure mein sirf 360° hai.
3. Orders of rotational symmetry: do jhandiyon wali line 2, do lines ka cross 4 (arc sirf angle dikhata hai), chhe nok wala star 6, teen taangon wali figure 3, plus ka nishaan 4, regular pentagon 5.

Kya angles hamesha sabse chhote angle ke multiples hote hain? Haan. Sabse chhote angle se do baar ghumana bhi mel khaata hai, isliye uske multiples angles of symmetry hain; aur koi aur angle of symmetry, kisi multiple jitna wapas ghumane pe, sabse chhote se bhi chhota mel khaata ghumaav de dega.

Sahi ya galat: Har figure ka 360° angle of symmetry hota hai: Sahi. Sabse chhota angle of symmetry natural number degrees mein ho to wo 360 ka factor hai: Sahi.

Circle ki symmetries (page 237)

Rotational symmetry wali aur cheezein: chhat ka pankha, cycle ka pahiya, kai phool, choodi, Ashoka Chakra aur kai rangoliyan.

Figure it Out: aur symmetry (pages 238–239)

1. Circle mein 30° ke 12 barabar sectors hain. i) 3 angles of symmetry ke liye har chautha sector bharo (ek sector, teen chhodo, aise hi), ya koi bhi pattern jo har 120° pe dohraaye. ii) 4 ke liye har teesra sector bharo, taaki pattern har 90° pe dohraaye. iii) Possible angles of symmetry ki ginti 1, 2, 3, 4, 6 aur 12 hai, yaani 12 ke factors: ek bhara sector 1 deta hai, do aamne-saamne ke sectors 2, har chautha sector 3, har teesra sector 4, ek chhod ke ek sector 6, aur saare sectors ek jaise 12.
2. Jaise, rectangle (2 lines of symmetry; angles 180° aur 360°) aur equilateral triangle (3 lines; angles 120°, 240° aur 360°). Plus ka nishaan aur regular hexagon bhi chalte hain.
3. a. Equilateral triangle: 3 lines aur 3 angles of symmetry. b. Sirf do barabar sides wala isosceles triangle: 1 line, rotational symmetry nahi. c. Aisa parallelogram jo rectangle ya rhombus na ho: koi line of symmetry nahi, angles of symmetry 180° aur 360°. d. Kite, ya barabar tirchhi sides wala trapezium: 1 line, rotational symmetry nahi.
4. Baaki angles of symmetry 60° ke multiples hain: 120°, 180°, 240°, 300° aur 360°.
5. Angles of symmetry sabse chhote angle ke multiples hain, aur 60° unme teesra hai, isliye sabse chhota 60° ÷ 3 = 20° hai. 60° se chhote angles 20° aur 40° hain.
6. a. Haan: 45, 360 ka factor hai (45 × 8 = 360), jaise 8 barabar radial arms wali figure. b. Nahi: 17, 360 ka factor nahi (17 × 21 = 357 aur 17 × 22 = 374).
7. a. Haan, bahari kinaare mein 3 lines of symmetry hain, har ek lambi side ke beech aur saamne ki chhoti side ke beech se. b. Haan, iske angles of rotational symmetry 120°, 240° aur 360° hain.
8. Regular polygons: triangle mein 3 lines, square mein 4, pentagon mein 5, hexagon mein 6, heptagon mein 7, octagon mein 8, nonagon mein 9 aur decagon mein 10. Sequence 3, 4, 5, 6, … hai, yaani 3 se shuru hone wale counting numbers.
9. Inke angles of symmetry ki ginti bhi wahi hai: 3, 4, 5, 6, …, kyunki n sides wala regular polygon 360° ÷ n ghumane pe apne aap pe aa jaata hai.
10. Koch snowflake sequence ki pehli shape, triangle, mein 3 lines aur 3 angles of symmetry hain; uske baad ki har shape mein 6 lines aur 6 angles of symmetry hain.
11. Ashoka Chakra mein 24 barabar teeliyan hain, isliye usme 24 lines of symmetry aur 24 angles of symmetry (15° ke multiples) hain.

Tiles se khelna (pages 239–240)

• a. Poore bhare upar-baayein chauthai hisse ko khadi laal line ke paar upar-daayein hisse mein reflect karo: har tile ka diagonal doosri taraf mud jaata hai aur uske dono rang jagah badal lete hain. Phir poore upar ke aadhe ko leti laal line ke paar neeche ke aadhe mein reflect karo. Poore design mein theek do laal lines hi lines of symmetry hain.
• b. 16 tiles se 4 × 4 square mein: 1 line ke liye daayan aadha baayein aadhe ka mirror image banao par upar aur neeche ki rows alag rakho; 2 lines ke liye ek 2 × 2 kona design karo aur use chaaron hisson mein reflect karo.
• c. Apne design: kone wale design ko reflect karne ki jagah ghuma ke rotational symmetry wale design bhi try karo.

Game (page 241)

Doosra khiladi hamesha jeet sakta hai. Jab bhi pehla khiladi koi line kheenche, doosra khiladi wo line kheenche jo uski grid ke centre ke chaaron taraf aadha chakkar ghumane wali copy ho. 6 by 6 grid mein koi line apni hi copy nahi ho sakti, aur koi square aur uski copy kabhi bagal mein nahi hote, isliye copy wali line hamesha khaali milti hai. Doosre khiladi ke paas hamesha chaal hoti hai, isliye pehla khiladi hi atakta hai.