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Tiling

Class 7 Maths · Chapter 14: Rachnaayein aur Tiling · NCERT Ganita Prakash section P2 6.2

Seekho

Tangram ek varg hai jo 7 tukdon (5 tribhuj, ek varg aur ek samaantar chaturbhuj) mein kata hai. Inse jaanvar, log aur naav ban sakte hain, aur har chitra ka kshetrafal varg jitna hi. Tiling matlab kisi kshetra ko aakaaron se poora dhakna, na khaali jagah, na overlap.

2 × 1 tiles (domino): har ek grid ke theek 2 varg dhakti hai. 4 × 6 grid (24 varg) dhak sakta hai; 5 × 7 (35 varg, odd) kabhi nahi. Grid tabhi dhakta hai jab vargon ki ginti even ho, yaani kam se kam ek bhuja even ho.

5 × 7 = 35 (odd) → impossible

Rang ki chaal: grid ko shatranj jaisa rango. Har domino ek kaala aur ek safed varg dhakta hai. Shatranj ke do ulte kone kaato: dono kone ek hi rang ke, to ek rang ke 30 aur doosre ke 32 bachte hain. 62 even hai, phir bhi 31 domino kabhi fit nahi honge! Yahi chaal batati hai ki 5 × 3 grid se kaunsa varg hataane par bacha kshetra nahi dhakega: agar ek rang ke 8 aur doosre ke 6 bachein, to asambhav.

Poora samtal dhakna: ek aakaar ki nakal farsh ko hamesha tak tabhi dhakti hai jab har kone ke angles milkar 360° hon. Varg (4 × 90°), samabahu tribhuj (6 × 60°) aur regular hexagon (3 × 120°, madhumakkhi ke chhatte jaisa) chalte hain. Regular pentagon (108°) nahi: 3 × 108° = 324° jagah chhodta hai. Aakaar milaana bhi chalta hai, jaise do octagon (135° har) aur ek varg (90°). Kalaakaar Escher ne jaanvaron ke aakaaron se samtal dhaka, aur madhumakkhiyan hexagon khaane banaati hain taaki jagah bekaar na ho.

135° + 135° + 90° = 360°

Explore: kaunse aakaar farsh dhakte hain?

  1. Aakaar sort karo: sirf ek aakaar ki nakal.

Yaad rakho

  • Tiling: na khaali na overlap; 2 × 1 tiles ko vargon ki even ginti chahiye.
  • Aakaar samtal tab dhakte hain jab har kone ke angles milkar 360°.

Quick check

Kya 3 × 5 grid theek 2 × 1 tiles se dhak sakta hai?

  1. Nahi: 15 varg, odd ginti
  2. Haan, 7 tiles se
  3. Haan, 8 tiles se
  4. Sirf tiles kaatkar
Answer

Nahi: 15 varg, odd ginti. Har tile 2 varg dhakti hai, to kul even hona chahiye.

Socho

Regular pentagon akele farsh kyun nahi dhak sakte?

Ek answer

Regular pentagon ka har angle 108°. Teen se 324° (36° ki jagah) aur chaar se 432° (overlap). Koi poori ginti theek 360° nahi banaati.

Practice

Teen levels (easy, medium, hard) ke naye sawaalon se practice karo, kuch hands-on, hints aur hal ke saath. Topic poora karne ke liye pichhle 5 mein se 4 sahi karo.