Class 6 Maths Chapter 8 NCERT Solutions Hinglish mein: Playing with Constructions

Ganita Prakash (Reprint 2026-27) ke Chapter 8, Playing with Constructions, ke har “Figure it Out” aur “Construct” kaam ke jawab, saath mein wo in-text sawal jo unse jude hain. Page numbers book se milte hain. Constructions notebook mein bante hain, isliye har jawab mein steps shabdon mein, kaunsi lambaiyan leni hain, aur kya dikhna chahiye, ye diya hai. Short notes: Chapter 8 notes.

P se 4 cm door points (pages 188–189)

Socho P se 4 cm door saare points. Wo kaise dikhenge? Wo P ke chaaron taraf ek gol curve mein jud jaate hain: circle. Banane ke liye compass ko ruler ke saath 4 cm kholo, needle ko P pe tika ke rakho aur pencil ko ek baar poora ghumao.

Circle pe ek point lo. Wo P se kitna door hai? Theek 4 cm, aur circle ka har doosra point bhi. P centre hai aur 4 cm radius hai.

Fig. 8.1 ka artwork dobara banana. Har circle compass se banao:

• Do chhoote hue circles: same radius, jaise 2 cm, aur centres 4 cm door.
• Circle ke andar circle: ek centre aur do radius, jaise 2 cm aur 1 cm.
• Teen overlap karte circles: same radius, centres ek line pe barabar doori pe, do radius se thoda kam door (jaise radius 2 cm aur centres 3 cm door).
• Chehra: sir ke liye bada circle, kaanon ke liye do chhote circles jinke centres upar baayein aur daayein hon, aankhon ke liye dot wale chhote circles, naak ke liye ek dot aur munh ke liye ek chhoti line.
• Cross karti lines: do barabar lines jo beech mein ek doosre ko kaatein, aur har ek ke neeche wale sire pe same radius ka ek chhota circle.

Haan, instruments se kaam aasaan hota hai: circles gol bante hain, barabar circles sach mein barabar hote hain, aur centres ke beech ki doori naap ke unhe theek jagah rakh sakte ho.

Construct: A Person (pages 190–191)

Figure ke do hisse hain: sir aur gardan (ek circle aur ek line), aur body (ek rectangle jiska upar wala hissa neeche jhukta curve hai).

1. Body bina upar wali line ke banao: neeche ki side aur do khadi sides, jaise 4 cm chaudi aur 4 cm oonchi.
2. Upar ka curve dono upar ke corners se guzar ke neeche jhukna chahiye, isliye compass ki nok body ke upar, body ki beech wali line pe, dono corners se barabar doori pe jaati hai.
3. Chaudai ke aadhe se bada radius fix karo aur beech wali line pe nok rakh ke try karo jab tak arc dono corners se na guzre. Jaise, nok ko corners ke beech ke point se 1.5 cm upar aur radius 2.5 cm rakhne pe arc dono corners se guzarta hai aur 1 cm jhukta hai.
4. Curve ke sabse neeche ke point se seedhi upar gardan kheencho, aur sir ek circle banao jiska centre usi beech wali line pe ho aur jo gardan ke upar tika ho.

Figure it Out: Wavy Wave (page 191)

1. Dono waves AB = 8 cm ko barabar baantti hain, isliye AX = 4 cm. Pehli wave AX pe half circle hai jiska centre AX ke beech mein hai, isliye compass ka radius 2 cm hai.
2. Koi bhi lambai chalegi agar radius AB ka chautha hissa ho. AB = 6 cm ke liye: AX = 3 cm aur radius 1.5 cm, nok A se 1.5 cm aur 4.5 cm pe. AB = 10 cm ke liye: radius 2.5 cm, nok A se 2.5 cm aur 7.5 cm pe.
3. Half circle se chhoti waves ke liye nok line se hat ke, har wave ki beech wali line pe jaati hai, aur dono waves mein same radius aur same doori rehti hai. AB = 8 cm pe: AX ka beech (A se 2 cm) mark karo, usse ek chhota perpendicular kheencho, aur nok us par AB se 1.5 cm neeche rakho. Ab 2.5 cm ka radius theek A aur X se guzarta hai; AB ke upar arc banao. Doosri wave ke liye nok XB ke beech se 1.5 cm upar rakho aur usi 2.5 cm radius se AB ke neeche X se B tak arc banao. Dono waves ek jaisi hain, aur har ek sirf 1 cm uthti hai.

Construct: Eyes (page 192, page 215 ke Hint A ke saath)

Aankh ka har curve ‘A Person’ ke upar wale hisse ki tarah banta hai: aankh ke do corners se guzarta arc.

1. Aankh ke liye ek halki supporting line kheencho, jaise 6 cm lambi, aur uska beech mark karo. Beech se uske saath 90° pe ek halki line kheencho.
2. Is beech wali line pe supporting line se 4 cm neeche B aur 4 cm upar A mark karo.
3. Needle B pe rakho, compass ko supporting line ke ek sire tak kholo (5 cm) aur ek sire se doosre sire tak upar wala curve banao.
4. Needle A pe rakho aur usi khulaav se neeche wala curve banao.
5. Beech mein pupil ke liye ek chhota bhara hua circle banao. Doosri aankh ke liye pehli ke bagal mein doosri supporting line kheench ke yahi dohrao.

A aur B beech wali line pe, supporting line se barabar doori pe hone chahiye, taaki dono curves mel khayein aur aankh symmetrical bane. A aur B line se jitne door, aankh utni chapti.

Apni pasand ka artwork banao. Ek idea (Extra): 3 cm radius ka circle banao. Wahi khulaav rakh ke needle circle pe kahin bhi rakho aur circle ke andar ek taraf se doosri taraf tak arc banao. Needle wahan le jao jahan arc circle se milta hai aur dohrao. Chhe arcs se chhe pankhudiyon wala phool banta hai.

Is square ka naam kaunsa nahi hai? (page 193)

1. PQSR. Square ke chaaron taraf ghoomne pe corners S, P, Q, R ke kram mein aate hain. SPQR, RSPQ aur QRSP sab isi kram mein hain, par PQSR mein Q se S side ke saath nahi, square ke aar-paar jaata hai.

Figure it Out: dot paper pe square aur rectangle (page 194)

1. Dots aise gino ki tasveer baayein-daayein aur upar-neeche ek jaisi rahe. Jaise, rectangle 4 spaces lamba aur 2 spaces ooncha banao, har square 1 space by 1 space, aur har square ko rectangle se 1 space baayein ya daayein aur 1 space upar ya neeche rakho. Tab har square apne corner ke paas same jagah baithta hai, isliye chaaron symmetrical tarah se rakhe jaate hain.

2. Sirf A square hai.

   • A: har side 4 spaces aage aur 4 spaces upar ya neeche jaati hai, isliye chaaron sides barabar hain. 4 aage pe 4 neeche jaane wali side dots ki row ke saath aadha right angle (45°) banati hai, isliye corner pe milne wali do sides 45° + 45° = 90° banati hain. A ek ghumaya hua square hai.
   • B: har side 3 aage aur 4 neeche jaati hai, isliye sides barabar hain, par har side aadhe right angle se zyada khadi hai aur upar ka angle 90° se kam hai. Square nahi.
   • C: har side 5 aage aur 4 neeche jaati hai, isliye sides phir barabar hain, par upar ka angle 90° se zyada hai. Square nahi.
   • D: chhoti sides 2 aage aur 1 neeche jaati hain, aur lambi sides 2 aage aur 4 neeche. Iske angles 90° hain, par saari sides barabar nahi, isliye ye ghumaya hua rectangle hai, square nahi.

   Socho: haan, grid pe har side ke steps gin ke bina instruments ke bata sakte ho. Agar har side ke steps same hain, bas ghume hue, to sides barabar hain. Agar ek side 2 aage aur 1 neeche jaati hai aur agli 1 peeche aur 2 neeche, to doosri side pehli ko chauthai chakkar ghumane se banti hai, isliye unke beech ka angle 90° hai.

3. Kuch figures jinke corners dots pe hain (spaces gin ke):

   • Ghumaya hua square: ek corner se 2 daayein aur 1 neeche, phir 1 baayein aur 2 neeche, phir 2 baayein aur 1 upar, phir 1 daayein aur 2 upar, wapas shuru pe.
   • Bada ghumaya hua square: 3 daayein aur 1 neeche, phir 1 baayein aur 3 neeche, phir 3 baayein aur 1 upar, phir 1 daayein aur 3 upar.
   • Ghumaya hua rectangle: 2 daayein aur 1 neeche, phir 2 baayein aur 4 neeche, phir 2 baayein aur 1 upar, phir 2 daayein aur 4 upar.

   Squares mein har side ke steps same hain, isliye saari sides barabar hain; rectangle mein aamne-saamne ki sides ke steps same hain. Har mod chauthai chakkar hai, isliye saare angles 90° hain. Ruler aur protractor se check karo.

6 cm side ka square banana (pages 195–196)

PS 6 cm lamba kyun hona chahiye? Square ki saari sides barabar hoti hain (property S1). PQ = 6 cm hai, isliye PS bhi 6 cm hona chahiye.

RS kitna lamba hai, aur ∠R aur ∠S kitne hain? RS = 6 cm, aur ∠R = ∠S = 90°. PQRS ki saari sides barabar aur saare angles 90° hain, isliye wo square hai. Compass abhi bhi 6 cm khula ho to R ek baar mein mark hota hai: needle Q pe rakho aur Q se guzarte perpendicular ko kaato.

Construct: rectangles (page 197)

1. 4 cm by 6 cm. AB = 6 cm kheencho. A aur B pe AB ke perpendicular kheencho. Compass 4 cm pe rakh ke A wale perpendicular pe D aur B wale pe C mark karo, aur DC jodo. R1 check: AB = DC = 6 cm aur AD = BC = 4 cm. R2 check: chaaron angles 90° naapte hain.
2. 2 cm by 10 cm. AB = 10 cm kheencho, A aur B pe perpendicular kheencho, AD = BC = 2 cm mark karo aur DC jodo. Phir se aamne-saamne ki sides barabar hain (10 cm aur 2 cm) aur har angle 90° hai.
3. Nahi. AB kheench ke A aur B pe right angles banao: A aur B se sides seedhe upar jaati hain. Agar AD ko BC se chhota karo, to line DC tirchhi ho jaati hai, aur D aur C pe angles 90° nahi rehte. Isliye jab chaaron angles 90° hon, aamne-saamne ki sides hamesha barabar hoti hain, aur figure rectangle hai.

Rectangles mein ek exploration (pages 197–199)

Rectangle ABCD banao jisme AB = 7 cm aur BC = 4 cm. X, AD pe chalta hai aur Y, BC pe.

X aur Y sabse paas kab hain? Jab wo A aur B se barabar door hon, yaani ek level pe. Tab XY = 7 cm, AB jitna. Isse paas wo kabhi nahi aa sakte, kyunki sides AD aur BC poori lambai mein 7 cm door hain.

Wo sabse door kab hain? Jab X, A pe aur Y, C pe ho, ya X, D pe aur Y, B pe. Tab XY ek diagonal hai, AC ya BD, lagbhag 8.1 cm lamba. Yaani sabse zyada doori diagonal ke barabar hai.

Hisaab rakhna. Teen columns ki table se kaam ho jaata hai. Tumhari naap inke aas-paas aani chahiye:

X ki A se doori	Y ki B se doori	XY ki lambai
5 mm	3 cm	lagbhag 7 cm 4 mm
1 cm	1 cm	7 cm
2 cm	4 cm	lagbhag 7 cm 3 mm

Jab X aur Y, A aur B se barabar door hon (5 mm aur 5 mm, 1 cm aur 1 cm, 1 cm 5 mm aur 1 cm 5 mm, wagairah):

1. XY hamesha 7 cm, AB ke barabar hai.
2. ABYX ek rectangle hai: AB aur XY dono 7 cm, AX aur BY barabar, aur saare angles 90°.

Rectangle todna (pages 199–201)

Do ek jaise squares. Rough diagram ACDF mein ABEF aur BCDE squares hain, isliye AF = AB = BE = FE aur BE = BC = CD = ED: saari chhoti lines barabar hain. Agar AF = 4 cm, to AC = 4 + 4 = 8 cm. Bina do baar naape banane ke liye:

1. Kisi bhi lambai ka AF kheencho.
2. A se AF ke perpendicular ek line kheencho.
3. Compass ko AF ki lambai jitna kholo. Needle A pe rakh ke perpendicular pe B mark karo; needle B pe rakh ke C mark karo.
4. F pe AF ka aur C pe AC ka perpendicular kheencho; ye D pe milte hain. Usi khulaav se F se FD pe E mark karo, aur BE jodo.

Teen ek jaise squares. Yahi karo, par AF ki lambai A se perpendicular pe teen baar copy karo. AF = 3 cm ho to rectangle 3 cm by 9 cm hai, aur chhoti side se 3 cm aur 6 cm pe lines use teen squares mein kaat deti hain.

Jo rectangles nahi kat sakte:

• do ek jaise squares mein: koi bhi rectangle jiski lambi side chhoti side ki 2 guna na ho, jaise 3 cm by 5 cm ya 4 cm by 6 cm;
• teen ek jaise squares mein: koi bhi rectangle jiski lambi side chhoti side ki 3 guna na ho, jaise 3 cm by 8 cm, ya 4 cm by 8 cm, jo do squares mein kat jaata hai par teen mein nahi.

Construct: squares, rectangles aur curves (pages 201–203)

1. A Square within a Rectangle. Rough diagram mein square dono lambi sides ko chhoota hai, isliye uski side 4 cm hai, rectangle ki chhoti side. Lambi side ka 8 − 4 = 4 cm bachta hai, jo barabar baanta jaata hai: har sire pe 2 cm. 8 cm by 4 cm ka rectangle banao, dono lambi sides pe har sire se 2 cm pe points mark karo, aur upar ke har point ko uske neeche wale point se jodo. Square ke corners rectangle ke corners se 2 cm door hain, aur uska centre har chhoti side se 4 cm aur har lambi side se 2 cm door hai, theek wahi jahan rectangle ka centre hai.
2. Falling Squares. 4 cm side ka pehla square banao. Uski upar wali side ko 4 cm daayein aur daayein wali side ko 4 cm upar badhao: ye doosre square ki do sides hain, jo pehle ke upar-daayein corner se shuru hota hai. Perpendicular aur compass se use poora karo, aur teesre square ke liye dohrao. 3 cm, 5 cm aur 7 cm wale version mein neeche 7 cm ke square se shuru karo, uske upar-daayein corner pe isi tarah 5 cm ka square banao, aur 5 cm wale ke upar-daayein corner pe 3 cm ka square.
3. Shadings. 8 cm side ka bada square lo, taaki chhote squares 2 cm ke hon. Sides pe har 2 cm pe nishaan lagao aur grid ki lines kheencho, par neeche-baayein 4 cm by 4 cm ke hisse ko ek hi square rehne do, uske andar koi line nahi. Upar ki do rows ke har chhote square mein uske upar-baayein corner se neeche-daayein corner tak diagonal kheencho aur uske upar wala triangle shade karo. Neeche-daayein 4 cm by 4 cm ke hisse mein teen chhote squares mein yahi karo aur uska neeche-baayein chhota square khaali chhodo.
4. Square with a Hole. Circle ka centre square ka centre hona chahiye. Dono diagonals halke se kheencho, ya aamne-saamne ki sides ke beech ke points jodo; ye centre pe kaatte hain. Needle wahan rakho aur side ke aadhe se kam radius ka circle banao, jaise 6 cm ke square mein 2 cm radius.
5. Square with more Holes. Ek bada square banao, jaise 8 cm, aur aamne-saamne ki sides ke beech ke points jod ke use chaar 4 cm ke squares mein kaato. Har chhote square ka centre isi tarah dhoondho aur same radius ke chaar circles banao, jaise 1.2 cm.
6. Square with Curves. 8 cm ka square banao. Har curve ek side ke do corners se guzarta hai aur square ke andar ubharta hai, isliye uska centre square ke bahar, us side ki beech wali line pe hai. Needle ko square ke bahar, side ke beech ke saamne 4 cm pe rakho (ye bagal mein bane ek kalpanik 8 cm ke square ka centre hai), compass ko us side ke ek corner tak kholo (lagbhag 5.7 cm) aur ek corner se doosre tak arc banao. Chaaron sides pe same doori aur same radius lo, taaki chaaron curves barabar ubhrein, har ek lagbhag 1.7 cm.

Diagonals ki exploration (pages 203–204)

Diagonals ki lambai compare karo. Ye barabar hain: PR = QS. Har rectangle mein jo tum banao, ise check karo.

Kya g aur h barabar hain? Kya c aur d barabar hain? Aam rectangle mein nahi. Barabar wale hain a = d = e = h, lambi sides ke paas ke chaar hisse, aur b = c = f = g, chhoti sides ke paas ke chaar hisse. Lambi sides ke paas ke hisse chhote hote hain, aur har corner pe dono hisse milkar 90° banate hain.

Rectangle kaisa banayein ki diagonal aamne-saamne ke angles ko barabar hisson mein baante? Chaaron sides barabar karo: square mein har hissa 45° hai.

Record karna. Do side lengths, aath angle hisse a se h tak, aur dono diagonals ki lambai wali table pattern jaldi dikha deti hai.

Jo general niyam bana sakte ho:

• Rectangle ke dono diagonals barabar hote hain.
• Aath angle hisse chaar-chaar barabar angles ke do groups mein aate hain: lambi sides ke paas wale aur chhoti sides ke paas wale.
• Har corner ke dono hisse milkar 90° hote hain, aur chhota hissa lambi side ke paas hota hai.
• Square mein aathon hisse 45° ke hote hain.

Kaise pakka karein ki niyam hamesha sach hain? Kuch rectangles naapne se sirf itna pata chalta hai ki unme ye sach hain; ek kaaran sab ke liye kaam karta hai. Rectangle ko PQ aur SR ke beech ke points se guzarti line pe modo: P, Q pe aata hai aur S, R pe, isliye diagonal PR theek QS pe aata hai. Dono mel khaate hain, isliye PR = QS, aur d = e, c = f, h = a aur g = b. Ab rectangle ko uske centre ke chaaron taraf aadha chakkar ghumao: P, R pe jaata hai aur Q, S pe, jisse d = h aur c = g dikhta hai. In dono se chaar-chaar barabar hisson ke do groups milte hain.

Solved examples mein (pages 205–210)

• ∠A ka doosra hissa (example 1). ∠A 90° hai aur ek hissa 60° hai, isliye doosra hissa, ∠DAC, 30° hai.
• D dhoondhne ke do tareeke. C pe BC ka perpendicular kheencho, kyunki rectangle ke saare angles 90° hote hain; ya compass se AD = BC mark karo, kyunki aamne-saamne ki sides barabar hoti hain.
• Circle line l se kahan milta hai (example 2). Wo point D ke chaaron taraf 7 cm radius wale circle pe hai, isliye D se 7 cm door hai, aur l pe bhi hai. Isliye wahi B hai.

Construct: angles aur diagonals se rectangle (page 211)

1. Diagonal corners ko 50° aur 40° mein baante. Kisi bhi lambai ka AB kheencho, jaise 5 cm, aur B pe AB ka perpendicular. A pe protractor se AB ke saath 50° ki line kheencho; ye perpendicular ko C pe milti hai. A pe AB ka perpendicular kheencho aur compass se AD = BC rakh ke us par D mark karo (ya C pe BC ka perpendicular kheencho). CD jodo. Ab ∠CAB = 50° aur ∠DAC = 40°. AB = 5 cm ho to BC lagbhag 6 cm aata hai.
2. 45° aur 45°. Yahi steps A se 45° ki line ke saath karo. Is baar BC AB ke barabar aata hai, isliye chaaron sides barabar hain: rectangle ek square hai.
3. Side 4 cm, diagonal 8 cm. DC = 4 cm aur C pe DC ke perpendicular line l kheencho. D ko centre aur 8 cm radius lekar arc kheencho jo l ko B pe kaate. D aur B pe perpendicular kheencho; ye A pe milte hain. ABCD rectangle hai, aur BC lagbhag 6.9 cm naapta hai.
4. Side 3 cm, diagonal 7 cm. Wahi steps: DC = 3 cm, C pe perpendicular l, D se 7 cm radius ka arc l ko B pe kaate, phir rectangle poora karo. BC lagbhag 6.3 cm naapta hai.

Diagonal hamesha side se lamba hona chahiye: D se DC se chhota arc kabhi line l tak nahi pahunchega.

Do points se barabar door points: House (pages 211–214)

Kya figure poori kar sakte ho? Base (BD, DE aur EC, har ek 5 cm, 1 cm by 2 cm ke darwaze ke saath) ke baad point A chahiye, jo B aur C dono se 5 cm door ho.

Kya B ke chaaron taraf ka circle madad karta hai? B se 5 cm door har point B ko centre maan ke bane 5 cm radius ke circle pe hai, isliye A kahin is circle pe hai: wo is par ka wo point hai jo C se bhi 5 cm door ho.

Jahan dono circles kaatte hain. C ko centre maan ke bhi 5 cm radius ka circle banao. BC ke upar jahan dono circles kaatte hain, wo point B se 5 cm aur C se 5 cm door hai, isliye wahi A hai. Circles BC ke neeche, ghar ke andar, ek aur point pe bhi kaatte hain, jo dono se 5 cm door hai, par chhat ke liye upar wala chahiye.

Kya do poore circles zaroori the? Nahi. Sirf A ke paas ke hisse kaam ke hain, isliye 5 cm radius ke do chhote arcs, ek B se aur ek C se, kaafi hain.

Aakhri curve. A, B aur C dono se 5 cm door hai, isliye A ko centre maan ke 5 cm radius ka arc dono se guzarta hai. Use B se C tak banao, aur ghar taiyaar hai.

Construct: bada ghar aur aur bhi (page 215)

1. Saari sides 7 cm wala ghar. DE = 7 cm aur D aur E pe perpendicular kheencho. Compass se un par 7 cm pe B aur C mark karo. B aur C se 7 cm radius ke arcs kheencho; ye A pe kaatte hain. AB aur AC jodo, aur A ko centre maan ke 7 cm radius ka arc B se C tak banao. Darwaza DE ke beech mein 1 cm by 2 cm hi reh sakta hai, ya thoda bada bana sakte ho.
2. House wale idea se artwork. House dikhata hai ki do points se barabar door point kaise dhoondhein: dono points se same radius ke arcs kheencho, aur wo us point pe kaatte hain jo chahiye. Isse har curve ke liye compass ki nok dhoondho:
   • A Person: body ke do upar ke corners (jaise 4 cm door) se 2.5 cm radius ke arcs kheencho jo body ke upar kaatein. Needle kaatne wale point pe rakh ke usi radius se ek corner se doosre tak curve banao.
   • Wavy Wave: A aur X se same radius, jaise 2.5 cm, ke arcs kheencho jo AB ke neeche kaatein, aur wahan se pehli wave banao. X aur B se arcs kheencho jo AB ke upar kaatein, aur wahan se doosri wave banao. 2 cm radius, yaani AX ka aadha, lene pe arcs line pe hi milte hain aur waves half circles ban jaati hain.
   • Eyes: aankh ke do corners se, jo 6 cm door hain, 5 cm radius ke arcs kheencho. Ye do points pe kaatte hain, aankh ke beech se 4 cm upar aur 4 cm neeche. Upar wale curve ke liye neeche wala point aur neeche wale curve ke liye upar wala point use karo.
3. Haan, 4 sides wali figure ki saari sides barabar ho sakti hain bina square hue (Hint B). AC = 5 cm kheencho, aur AB = 5 cm aise angle pe jo 90° na ho, jaise 60°. Chautha corner D, B aur C dono se 5 cm door hona chahiye, isliye B aur C se 5 cm radius ke arcs kheencho: ye A pe aur ek aur point D pe kaatte hain. BD aur CD jodo. ABDC ki chaaron sides 5 cm hain, par uske angles 60° aur 120° hain, 90° nahi, isliye wo square nahi hai. (Extra: aisi figure ko rhombus kehte hain.)