Class 7 Maths · Chapter 14
Short answer: Class 7 Maths Ganita Prakash Chapter 14 "Constructions and Tilings" me ruler-compass se exact geometric constructions (perpendicular bisector, angle bisector, standard angles, equilateral triangle) aur tessellation/tiling ka concept sikhaya jaata hai — yani konse shapes bina gap ya overlap ke surface cover kar sakte hain.
Chapter 14 poore Class 7 Maths ki practical geometry ki neev hai — yaha ruler aur compass se exact constructions (perpendicular bisector, angle bisector, standard angles, triangles) sikhne ke saath-saath tiling/tessellation ka fascinating concept bhi cover hota hai, jisme dekha jaata hai ki konse geometric shapes bina gap ya overlap ke poori surface cover kar sakte hain. Ye NCERT Class 7 Maths Ganita Prakash chapter wise solutions series ka ek important part hai jo Constructions and Tilings class 7 practical geometry ke saare core concepts ko step-by-step explain karta hai.
Chapter 14 Summary — 5 Minute Revision
Constructions and Tilings chapter do bade concepts jodta hai: pehla, ruler aur compass se precise geometric constructions (perpendicular bisector, angle bisector, 60°/90°/120° angles, equilateral triangle) jisme har step exact geometric properties (equal radii = equal distances) par based hota hai, na ki approximate measurement par. Doosra, tiling ya tessellation ka idea — jaha sikhte hain ki equilateral triangle, square aur regular hexagon jaise shapes akele poori surface bina gap ya overlap ke cover kar sakte hain, jabki pentagon jaisi shapes nahi kar paatin, kyunki unka interior angle 360° ko exactly divide nahi karta. Ye dono concepts real-life applications jaise floor tiling, pattern design aur architecture me directly use hote hain, isliye CBSE exams me diagram-based aur step-marking construction questions ke roop me regularly poochhe jaate hain.In-Text Questions — Solutions
Figure it out: Ek line segment ka perpendicular bisector construct karke check karo ki woh segment ko exactly do equal parts me divide karta hai ya nahi.
Line segment AB draw karke A aur B se equal radius (AB ki half se zyada) ke arcs lo jo do points par cross karein. In points ko milane wali line AB ko M par cut karegi, jaha AM = MB. Scale se measure karke verify kar sakte ho ki dono parts exactly equal hain — yehi perpendicular bisector ki property hai.Ek angle ko compass se bisect karo aur verify karo protractor se ki dono halves equal hain.
Angle ke vertex O se ek arc lo jo dono rays ko cut kare (points A, B). Phir A aur B se equal radius ke arcs lo jo ek dusre ko point C par cross karein. O se C tak ray hi original angle ko do equal parts me bisect karti hai. Protractor se check karne par dono halves same degree measure denge.Exercise Questions — Solutions (Q1–Q6)
Q1. Ruler aur compass ka use karke, ek line segment PQ = 6.4 cm ka perpendicular bisector construct karo.
Steps:
- Line segment PQ = 6.4 cm draw karo.
- P ko center bana kar, PQ ki half se zyada radius lekar (jaise 4.5 cm) ek arc line ke dono taraf draw karo.
- Q ko center bana kar wahi radius lekar phir se arc draw karo, jisse pehle wale arcs ko do jagah cut kare — points A (upar) aur B (neeche).
- A aur B ko milao — ye line hi PQ ka perpendicular bisector hai, jo PQ ko M par 90° pe cut karti hai.
PM = MQ = 3.2 cm
Q2. Ek point P se, ek di gayi line l ke bahar, l par perpendicular draw karo (compass method se).
Steps:
- P ko center bana kar ek arc draw karo jo line l ko do points A aur B par cut kare.
- Ab A aur B ko centers bana kar, radius A B se zyada rakh kar, dono taraf arcs draw karo jo ek dusre ko point Q par cut karein (l ke opposite side pe P ke).
- P aur Q ko milao — line PQ hi l par perpendicular hai.
Q3. 60° ka angle construct karo compass se, bina protractor use kiye.
Steps:
- Ray OA draw karo.
- O ko center bana kar koi bhi radius se arc draw karo jo OA ko point B par cut kare.
- B ko center bana kar wahi radius rakh kar arc draw karo jo pehle arc ko point C par cut kare.
- O se hote hue C tak ray draw karo — angle AOC = 60° ban gaya.
∠AOC = 60°
Q4. Ek equilateral triangle ki side 5 cm hai. Iska construction karo aur batao ki uske teeno angles kitne degree ke honge.
Steps:
- Base AB = 5 cm draw karo.
- A ko center bana kar 5 cm radius se arc draw karo.
- B ko center bana kar 5 cm radius se dusra arc draw karo, jo pehle arc ko point C par cut kare.
- A-C aur B-C milao — triangle ABC equilateral ban gaya.
Equilateral triangle ke saare sides equal hote hain, isliye saare angles bhi equal honge:
3x = 180° ⟹ x = 60°
Teeno angles = 60° each
Q5. Ek square tile ka side 4 cm hai. Agar is tile se ek room ka floor (length 4 m, width 3 m) cover karna ho, to kitni tiles lagengi? (Tiling ka concept use karo)
Room ka area = length × width = 4 m × 3 m = 12 m² = 120000 cm²
Ek tile ka area = 4 × 4 = 16 cm²
Tiles ki sankhya = 120000 ÷ 16 = 7500 tiles
Square tiling ek regular tessellation hai kyunki squares gaps ya overlap ke bina poori surface cover kar dete hain — yehi wajah hai floor pe square tiles perfectly fit hoti hain.
Q6. Batao ki regular hexagon (6-sided shape) tiling (tessellation) kar sakta hai ya nahi. Reasoning do angle sum ke through.
Regular hexagon ka ek internal angle:
Interior angle = (n − 2) × 180° / n = (6−2) × 180° / 6 = 720°/6 = 120°
Ek point par jitne hexagons milenge, unka total angle 360° hona chahiye tiling ke liye (koi gap ya overlap nahi):
360° ÷ 120° = 3
Chunki 3 ek whole number hai, isliye haan, regular hexagon perfectly tile kar sakta hai — 3 hexagons ek vertex par bina gap ke milte hain (jaise honeycomb pattern).
Important Equations — Ek Nazar Me
| Concept | Rule / Formula |
|---|---|
| Perpendicular bisector | Har point equidistant hota hai dono endpoints se: PM = MQ |
| Angle bisector | Original angle ko do equal halves me divide karta hai |
| Equilateral triangle angle | Har angle = 60° (sum = 180°, teeno equal) |
| Regular polygon interior angle | Interior angle = (n − 2) × 180° / n |
| Tiling (tessellation) condition | 360° ÷ interior angle = whole number |
| Shapes jo akele tile karte hain | Equilateral triangle (60°), Square (90°), Regular hexagon (120°) |
↔ Table ko side me swipe karein
Common Mistakes — Yahan Marks Kat te Hain
- Perpendicular bisector banate waqt radius line segment ki half length se kam le lena — isse arcs cross hi nahi karte aur construction fail ho jaata hai.
- Angle construction me compass ki opening (radius) beech me change kar dena — ek hi step ke andar radius change nahi karna chahiye, warna angle galat ban jaata hai.
- Construction ke baad arcs/construction lines erase kar dena — CBSE marking scheme me construction marks tabhi milte hain jab saari arcs, lines properly dikhayi ho.
- Tiling questions me interior angle formula (n−2)×180°/n galat apply karna — 'n' ko sides ki jagah kuch aur number samajh lena.
- 360° ko interior angle se divide karke non-integer aane par bhi 'tile ho sakta hai' likh dena — yaad rakho sirf whole number result par hi single shape se tiling possible hoti hai.
- Equilateral triangle construct karte waqt dono arcs ka radius alag-alag le lena — dono arcs ka radius hamesha side length ke barabar aur same hona chahiye.
Board-Style Important Questions
- School/board-level practice papers me is topic se ruler-compass construction (perpendicular bisector, angle bisector) ke step-marking wale questions frequently poochhe jaate hain.
- Tessellation identify karne wale questions common hain — jaise 'kaunsa regular polygon akela tiling kar sakta hai aur kyun' with angle-sum reasoning.
- Equilateral triangle ya specific angle (60°/90°/120°) construct karne ke steps likhne wale questions practice worksheets me aate hain.
- Real-life tiling application questions (floor/wall tiles ka area aur count nikalna) combined concept ke roop me poochhe jaate hain.
- Interior angle formula use karke kisi regular polygon ka angle nikalna aur uski tiling-capability judge karna — conceptual reasoning based questions.
Aksar Poochhe Jaane Wale Sawaal
Class 7 Maths Chapter 14 Constructions and Tilings me kya-kya topics cover hote hain?
Is chapter me ruler-compass se practical geometry constructions sikhaye jaate hain — perpendicular bisector, angle bisector, standard angles (60°, 90°, 120°), triangles construct karna — aur dusre half me tiling/tessellation ka concept, jisme dekha jaata hai konse shapes (square, triangle, hexagon) surface ko bina gap/overlap ke cover kar sakte hain.
Perpendicular bisector construction me radius kitna lena chahiye?
Radius hamesha line segment ki half length se zyada lena chahiye, warna arcs cross nahi karenge aur bisector point nahi milega. Jaise agar PQ = 6 cm hai, to radius kam se kam 3.5 cm ya usse zyada lena best rehta hai — accuracy ke liye 4 cm ya usse thoda upar lena safe hai.
Tessellation/tiling ka matlab kya hai aur konse shapes tile kar sakte hain?
Tessellation ka matlab hai ek ya zyada shapes ko is tarah repeat karna ki poori surface bina kisi gap ya overlap ke cover ho jaaye. Regular polygons me sirf equilateral triangle (60°), square (90°) aur regular hexagon (120°) akele tile kar sakte hain — kyunki inke interior angles 360° ko exactly divide karte hain (360÷60=6, 360÷90=4, 360÷120=3).
60° ka angle bina protractor ke kaise banaye?
Ek ray draw karo, us par center se ek arc lo jo ray ko cut kare, phir usi radius se us cut point se dusra arc lo jo pehle arc ko cross kare — dono intersection points ko jodkar 60° ka angle mil jaata hai. Ye method equilateral triangle ki property (saare angles 60°) pe based hai.
Kya square aur regular pentagon dono tiling kar sakte hain?
Square tiling kar sakta hai kyunki uska interior angle 90° hai aur 360÷90=4 (whole number). Regular pentagon tiling nahi kar sakta kyunki uska interior angle 108° hai aur 360÷108 = 3.33 (whole number nahi) — isliye pentagons ke beech gaps reh jaate hain, poori surface cover nahi hoti.
Constructions me compass ka use kyu zaroori hai, ruler se seedha kyu nahi maap lete?
Compass se banaye gaye constructions geometrically exact hote hain — inme koi measurement error nahi aata jaisa ki protractor ya scale se andaza lagane me ho sakta hai. NCERT ki approach hamesha 'exact construction' sikhati hai jisme sirf ruler (unmarked) aur compass allowed hote hain, taaki logical geometric properties (jaise equal radii = equal distances) use ho sakein, na ki approximate measurement.
Class 7 Maths — Saare Chapters
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