Class 7 Maths Ch 7 NCERT Solutions Hinglish mein

Ye Ganita Prakash, yaani Class 7 Maths ki kitaab Part 1 (Reprint 2026-27), ke Chapter 7, A Tale of Three Intersecting Lines, ke har sawaal ke jawaab hain: text ke sawaal, har Construct task, chhahon Figure it Out sets, Math Talk aur Try This ke sawaal, aur makdi waali paheli. Page numbers kitaab ke hisaab se hain (pages 147–172). Constructions kadam ba kadam samjhaaye gaye hain; inhe khud ruler, compass, protractor aur set square se kheencho. Chhote notes: Chapter 7 ke notes.

Triangles (page 147)

Teeno vertices ek seedhi line par hon to kya hota hai? Triangle nahi banta: teeno ‘sides’ ek hi line par padi rehti hain aur koi jagah nahi gherti. Tab ek side baaki do ke jod ke baraabar hoti hai, aur triangle inequality theek isi case ko rokti hai.

Chapter 7 ka mind map: triangle banaana, triangle inequality, di gayi sides aur angles, angle sum property, altitudes aur triangles ke types — Poora chapter ek page pe

7.1 Equilateral Triangles (pages 147–148)

Har side 4 cm waala triangle banaao. Kaunse tools lagaaye? Kya sirf nishaan waale ruler se ban sakta hai? Sirf ruler se ban sakta hai, par baar baar koshish karke: AB = 4 cm kheenchne ke baad hum A se 4 cm door point C lagaate rehte hain jab tak wo B se bhi 4 cm door na ho. Compass se ye seedha hota hai: AB = 4 cm kheencho, A aur B dono se 4 cm radius ke arcs kheencho, aur unke milne ke point C ko A aur B se jodo.

AC aur BC dono 4 cm kyun hain? C, A centre aur 4 cm radius waale arc par hai, isliye AC = 4 cm, aur B centre aur 4 cm radius waale arc par bhi, isliye BC = 4 cm.

Do arcs se 4 cm side waala equilateral triangle banaana, A aur B dono se 4 cm radius — Equilateral triangle banaana

7.2 Teen sides se triangle (pages 148–150)

4 cm, 5 cm aur 6 cm sides waala triangle banaao. AB = 4 cm kheencho; A se 5 cm radius ka arc; B se 6 cm radius ka arc jo use C par kaate; AC aur BC jodo.

4, 5 aur 6 cm sides waala triangle banaana: base, A se 5 cm ka arc aur B se 6 cm ka arc, jo C par milte hain — Teen sides: 4 cm, 5 cm aur 6 cm

Construct (a) 4, 4, 6; (b) 3, 4, 5; (c) 1, 5, 5; (d) 4, 6, 8; (e) 3.5, 3.5, 3.5. Har baar yahi steps lagaao, sabse lambi side ko base lekar. (a) aur (c) isosceles hain, aur (c) lamba aur patla hai; (b) mein 3 cm aur 4 cm sides ke beech right angle nikalta hai; (d) scalene hai; (e) equilateral hai.

Construct task ke paanchon triangles scale ke saath — Construct: paanch triangles

Figure it Out (pages 150–151)

1. Circle se isosceles triangles. Centre O ko circle ke kinhi do points P aur Q se jodo: OP aur OQ radii hain, isliye ∆OPQ isosceles hai. Points ka har aisa jodaa ek naya triangle deta hai.

2. Ek jaise size ke do (ya teen) circles. Pehle chitra mein har circle doosre ke centre se guzarta hai, isliye AB radius ke baraabar hai. A, B aur circles ke milne ka koi bhi point equilateral triangle banaate hain, kyunki teeno sides radii hain. B centre waale circle ke kisi aur point C ke liye ∆ABC isosceles hai (BA = BC), aur A centre waale circle ke points ke liye bhi yahi. Doosre chitra mein A, B aur C mein se har ek baaki do circles par hai, isliye ∆ABC equilateral hai, aur har centre apne circle ke do points ke saath phir isosceles triangles banaata hai.

Circle ke centre aur us par ke do points se bane isosceles triangles, aur do baraabar circles se bane equilateral triangles — Circles se isosceles aur equilateral triangles

Kya har lambaai se triangle banta hai? (page 151)

3 cm, 4 cm aur 8 cm; 2 cm, 3 cm aur 6 cm. Dono se nahi banta: arcs kabhi nahi milte. Aur namumkin sets: 1, 2, 5; 2, 2, 6; 4, 5, 10. Pattern: har set mein sabse lambi lambaai baaki do ke jod ke baraabar ya usse zyaada hai.

Triangle Inequality (pages 151–153)

10 cm, 15 cm aur 30 cm? Triangle nahi banta: seedha raasta CA = 30 cm ghoomkar jaane waale raaste CB + BA = 25 cm se lamba ho jaata, jo namumkin hai.

10, 15 aur 30 sides waala kachcha triangle, har side jaanchi gayi: 30, 10 + 15 se kam nahi, isliye triangle nahi banta — Triangle inequality: 10, 15 aur 30?

3 cm, 3 cm aur 7 cm? Nahi: 7 > 3 + 3 = 6. Banaakar dekho to 7 cm base ke siron se kheenche 3 cm radius ke do arcs nahi milte.

Math Talk: kya lambaaiyon ko aisa jamaaya ja sakta hai ki har seedha raasta chhota ho? Nahi. 30 kisi bhi side ko do, wo baaki do sides ke jod (10 + 15 = 25) se lambi hi rahegi, isliye ek tulna hamesha fail hogi.

Figure it Out (page 154)

1. Haan, bina banaaye: 8 > 3 + 4 = 7 aur 6 > 2 + 3 = 5, isliye har set mein ek lambaai baaki do ke jod se lambi hai.

2. (a) 10, 10, 25 km: nahi, 25 > 20. (b) 5, 10, 20 mm: nahi, 20 > 15. (c) 12, 20, 40 cm: nahi, 40 > 32.

3 (Try This). Kya kam se kam do tulnaayein hamesha sahi aayengi? Haan. Do chhoti lambaaiyan sabse lambi se badi nahi hoti, isliye har ek aise jod se kam hai jisme sabse lambi shaamil hai. Jaise, 7, 10, 15 mein 7 < 25 aur 10 < 22 turant dikhta hai. Asal jaanch sirf sabse lambi ki karni padti hai, isliye sabse lambi lambaai ko baaki do ke jod se milaao.

Circles ke construction ki kalpana (pages 155–156)

4 cm, 5 cm aur 8 cm. 8 < 4 + 5 = 9, isliye set triangle inequality poori karta hai. Baaki do tulnaayein jaanchne ki zaroorat nahi, kyunki 4 aur 5 dono akele 8 se hi kam hain. 8 cm base AB kheenchne par A ke charon taraf ka 4 cm circle X tak pahunchta hai, jahan BX = 4 cm hai, jo B ke charon taraf ke 5 cm circle ke andar hai, isliye circles do points par kaatte hain aur triangle banta hai.

Figure it Out (page 156)

1. (a) 2, 2, 5: nahi, 5 > 4. (b) 3, 4, 6: haan, 6 < 7. (c) 2, 4, 8: nahi, 8 > 6. (d) 5, 5, 8: haan, 8 < 10. (e) 10, 20, 25: haan, 25 < 30. (f) 10, 20, 35: nahi, 35 > 30. (g) 24, 26, 28: haan, 28 < 50.

Do circles ke teen cases (pages 157–159)

Triangle inequality na maanne waale sets jinmein circles (a) ek point par chhoote hain: 2, 3, 5; 4, 4, 8; 1, 6, 7 (do chhoti lambaaiyon ka jod sabse lambi ke baraabar). (b) milte hi nahi: 2, 3, 6; 1, 1, 5; 3, 4, 9 (do chhoti lambaaiyon ka jod kam).

Poori vidhi: teeno lambaaiyon ko kram mein rakho; do chhoti lambaaiyon ko jodo; jod sabse lambi se zyaada ho to triangle banta hai, warna nahi.

Sabse lambi side ke siron se bane do circles chhoote hain, nahi milte ya kaatte hain, is par ki radii ka jod AB ke baraabar, kam ya zyaada hai — Kya dono circles milte hain?

Figure it Out (page 159)

1. (a) 1, 100, 100: banta hai, 100 < 101. (b) 3, 6, 9: nahi banta, 9 = 3 + 6 (circles bas chhoote hain). (c) 1, 1, 5: nahi banta, 5 > 2. (d) 5, 10, 12: banta hai, 12 < 15.

2. Haan, 50 < 50 + 50. Aam taur par kisi bhi side ki lambaai a ke liye equilateral triangle banta hai, kyunki a < a + a hamesha sach hai.

3. Teesri side ki paanch sambhav values: (a) 1 aur 100 ke saath: 99.5, 99.9, 100, 100.4, 100.9. (b) 5 aur 5 ke saath: 1, 3, 5, 7, 9.5. (c) 3 aur 7 ke saath: 4.5, 5, 6, 8, 9.9.

Try This: teesri side ki saari sambhav values. Di gayi do sides ke fark aur jod ke beech, dono ko chhodkar: (a) 99 aur 101 ke beech, (b) 0 aur 10 ke beech, (c) 4 aur 10 ke beech.

Number lines par sambhav teesri sides: 99 aur 101 ke beech, 0 aur 10 ke beech, aur 4 aur 10 ke beech — Sambhav teesri sides

7.3 Do sides aur beech ka angle (pages 160–161)

AB = 5 cm, AC = 4 cm aur ∠A = 45° waala ∆ABC banaao. AB = 5 cm kheencho, A par 45° ki arm banaao, us par C aisa lagaao ki AC = 4 cm ho, aur BC jodo.

AB = 5 cm, angle A = 45° aur AC = 4 cm waala triangle ABC banaana — Do sides aur beech ka angle

Figure it Out (page 161)

1. Har ek mein ek side kheencho, uske ek sire par angle banaao, aur nayi arm par doosri side naapo: (a) 7 cm base, ek sire par 75°, arm par 3 cm; (b) 6 cm base, 25°, 3 cm; (c) 8 cm base, 120°, 3 cm (obtuse-angled triangle).

Math Talk: kya yahan triangle kabhi namumkin hota hai? Sirf tab jab included angle 180° ya usse zyaada ho, kyunki tab dono sides ek line par padti hain ya ek doosre se door khulti hain. 180° se kam kisi bhi angle ke liye construction hamesha triangle banaata hai.

Do angles aur beech ki side (pages 161–163)

AB = 5 cm, ∠A = 45° aur ∠B = 80° waala ∆ABC banaao. AB kheencho, A par 45° aur B par 80° banaao; arms C par milti hain, aur ∠C 55° naapta hai.

AB = 5 cm, angle A = 45° aur angle B = 80° waala triangle ABC banaana — Do angles aur beech ki side

Figure it Out (page 162)

1. (a) 75°, 5 cm, 75°: teesra angle 30° waala isosceles triangle. (b) 25°, 3 cm, 60°: teesra angle 95°. (c) 120°, 6 cm, 30°: teesra angle 30°, ek isosceles obtuse-angled triangle.

Kya triangle hamesha banta hai? Udaaharan jahan nahi banta: 90° aur 90°; 100° aur 80°; 120° aur 70°; 40° aur 150°.

A par 40° ke saath: (a) 140° ya usse bada koi bhi ∠B, jaise 150°, arms ko milne nahi deta. (b) Aisa sabse chhota ∠B 140° hai: tab B se nikli arm pehli arm ke parallel hai, kyunki transversal AB ki ek hi taraf ke interior angles ka jod 180° hota hai. Lambaai AB ka isme koi haath nahi.

Base AB aur A par 40°: 140° par B se nikli arm l ke parallel hai aur use kabhi nahi milti — Arms kab kabhi nahi milti?

Figure it Out (page 163)

1. Aisa doosra angle jisse triangle bane ya na bane:

(a) 30°: 150° se kam kisi bhi angle ke saath banta hai, jaise 50° ya 90°; 150° ya usse zyaada ke saath nahi, jaise 150° ya 170°.
(b) 70°: 110° se kam ke saath banta hai, jaise 60° ya 80°; 110° ya usse zyaada ke saath nahi, jaise 120° ya 140°.
(c) 54°: 126° se kam ke saath banta hai, jaise 64° ya 90°; 126° ya usse zyaada ke saath nahi, jaise 134° ya 154°.
(d) 144°: 36° se kam ke saath banta hai, jaise 20° ya 35°; 36° ya usse zyaada ke saath nahi, jaise 36° ya 90°.

2. (a) 35°, 150°: nahi ho sakte, jod 185°. (b) 70°, 30°: ho sakte hain, jod 100°. (c) 90°, 85°: ho sakte hain, jod 175°. (d) 50°, 150°: nahi ho sakte, jod 200°.

Niyam: do angles kisi triangle ke angles theek tab ho sakte hain jab unka jod 180° se kam ho.

Teesra angle (page 164)

5 cm base par 60° aur 70°, phir 7 cm base par: teesra angle dono baar 50° hai; ye side ki lambaai par nirbhar nahi. Math Talk, hisaab rakhna: kai triangles ke liye do diye gaye angles aur naapa gaya teesra angle likho, har triangle ki ek row, aur har row ko jodo; saare jod lagbhag 180° aate hain. Sambandh: teesra angle 180° mein se baaki do ka jod ghataane par milta hai. ∠B = 50°, ∠C = 70° waala ∆ABC: A se BC ke parallel line XY alternate angles ∠XAB = 50° aur ∠YAC = 70° deti hai, isliye ∠BAC = 180° − 120° = 60°.

Triangle ABC aur A se BC ke parallel line XY: alternate angles dikhaate hain ki teeno angles ka jod 180° hai — Angles ka jod 180° kyun hai

Figure it Out (page 165)

1. (a) 36°, 72°: teesra angle 72°. (b) 150°, 15°: 15°. (c) 90°, 30°: 60°. (d) 75°, 45°: 60°. Har ek mein teesre vertex se saamne ki side ke parallel line diye gaye do angles ko alternate angles bana deti hai, aur us vertex ke teeno angles milkar 180° hote hain.

2. Nahi, teen 70° angles ka jod 210° hai. Do angles 70° hon to teesra 40° hai. Teeno baraabar hon to har ek 180° ÷ 3 = 60° hai.

3. ∠B + ∠C = 180° − 50° = 130°, aur dono baraabar hain, isliye ∠B = ∠C = 65°.

Angle sum property se nikaale gaye teesre angles — Teesra angle nikaalna

Angle Sum Property (pages 165–166)

Kaagaz ka triangle modna. Upar ke vertex ko base par neeche modo, phir baaki do kone andar modkar use milaao. Teeno angles seedhi base par ek ke baad ek fit ho jaate hain, isliye unka jod 180° hai.

Kaagaz ka triangle jiske teeno kone modkar base par milaaye gaye, angles ka jod 180° dikhaate hue — Modkar angle sum jaanchna

Exterior Angles (page 167)

∠A = 50° aur ∠B = 60° ho to ∠ACD: ∠ACB = 70°, isliye ∠ACD = 180° − 70° = 110°. Sambandh: ∠A + ∠B + ∠ACB = 180° aur ∠ACD + ∠ACB = 180° hai, isliye exterior angle ∠ACD hamesha ∠A + ∠B ke baraabar hai.

Triangle ABC aur BC ko D tak badhaaya gaya: 110° ka exterior angle 50° + 60° ke baraabar hai — Exterior angle

7.4 Altitudes (pages 167–169)

Height se kya matlab hai? Neeche se upar tak seedhi doori, zameen se right angle par naapi gayi. Triangle mein kisi vertex ki height saamne ki side se uski perpendicular doori hai.

B par angle obtuse ho to A se BC tak altitude: CB ko B ke aage badhaao aur A se is badhi line par perpendicular giraao; altitude triangle ke bahar padta hai.

Kaagaz modna. Aise modo ki base khud par aa jaaye aur mod upar ke vertex se guzre. Tab mod base ke saath jo do angles banaata hai ve ekdum baraabar hain aur milkar seedha angle banaate hain, isliye har ek 90° hai: mod base par perpendicular hai.

Ruler aur BC ke saath khiskte set square se A se altitude kheenchna — Set square se altitude kheenchna

Aisa triangle jisme ek side altitude bhi ho: right-angled triangle; right angle banaane waali do sides mein se har ek doosri par perpendicular hai.

Altitudes: acute triangle mein andar, aur obtuse triangle mein bahar, jahan BC aage badhaayi gayi hai — Triangle ke altitudes

7.5 Triangles ke types (page 170)

Koi aur type? Haan: obtuse angle waale triangles, jaise is chapter ke 120° waale triangles aur 140° waala triangle TRY, aur aise triangles jinke saare angles acute hon. Inse angles ke hisaab se baantna banta hai. Acute-angled triangle ko ek acute angle waala triangle kyun na kahein? Har triangle mein kam se kam do acute angles hote hain, isliye wo paribhasha har triangle par laagu ho jaati; acute-angled triangle ke teeno angles acute hone chahiye.

Sides ke hisaab se triangles (equilateral, isosceles, scalene) aur angles ke hisaab se (acute, right, obtuse) — Triangles ke types

Figure it Out (pages 170–171)

1. BC = 5 cm, AB = 6 cm, CA = 5 cm. BC = 5 cm kheencho, B se 6 cm radius ka arc aur C se 5 cm ka, jo A par milte hain. Triangle isosceles hai (CA = CB) aur iske saare angles acute hain, isliye A se altitude BC ke andar girta hai; naapne par ye lagbhag 4.8 cm hai, aur BC ko B se lagbhag 3.6 cm par milta hai.

2. RY = 4 cm, TR = 7 cm, ∠R = 140°. RY = 4 cm kheencho, R par RY ke saath 140° banaati arm kheencho, aur us par 7 cm par T lagaao; TY jodo. R par angle obtuse hai, isliye YR ko R ke aage badhaao aur set square se T se badhi line par perpendicular giraao; ye triangle ke bahar hai, lagbhag 4.5 cm hai, aur badhi line ko R se lagbhag 5.4 cm aage milta hai.

3. ∠B = 90°, AC = 5 cm. Anginat triangles. AC ko base lo; ∠A 0° aur 90° ke beech koi bhi angle ho sakta hai, aur ∠C = 90° − ∠A; har choice ek alag right triangle deti hai.

∠B = 90° aur AC = 5 cm waale kai right triangles, unke vertices B, AC par bane semicircle par — Sawaal 3

4. Equilateral triangle right-angled ya obtuse-angled koi nahi ho sakta, kyunki iske saare angles 60° hain. Isosceles triangle dono ho sakta hai: 90°, 45°, 45° angles ke saath right-angled, aur 120°, 30°, 30° jaise angles ke saath obtuse-angled.

Kaunse mel mumkin hain: equilateral triangle right ya obtuse nahi ho sakta, par isosceles dono ho sakta hai — Sawaal 4

Dabbe mein sabse chhota raasta (page 172)

Makdi ka sabse chhota raasta. Jin do faces par makdi ko chalna hai (jaise saamne waala aur upar waala), unhe khol kar seedha bichha do taaki ve ek rectangle ban jaayein. Is chapte net par makdi ke kone se door ke kone tak seedhi line sabse chhota raasta hai. Waapas modne par ye raasta saajhi edge ko kisi kone par nahi, uske beech ke kisi point par paar karta hai. Faces ke alag jodon se alag lambaaiyan milti hain, isliye net par har seedhi line naapo aur sabse chhoti chuno.

Makdi ka raasta: do faces khol kar bichhaaye gaye, makdi se door ke kone tak seedhi line — Dabbe mein sabse chhota raasta

Chhote notes aur zyaada practice topic pages par hain: triangle kaise banayein, triangles ke types aur triangle inequality aur angle sum property.

NCERT Solutions Class 7 Maths Chapter 7: A Tale of Three Intersecting Lines (Hinglish)