English

Jyamitiya Rachnaayein

Class 7 Maths · Chapter 14: Rachnaayein aur Tiling · NCERT Ganita Prakash section P2 6.1

Seekho

Sirf compass aur seedhi patti se hairaan karne wali kai cheezein ban sakti hain. Zoya ek "aankh" banaati hai: khand ke siron X aur Y se guzarte do chaap, ek XY ke upar ke kendra se aur ek neeche se, dono X aur Y se barabar door. Aise saare kendra ek khaas line par hote hain.

Perpendicular bisector: compass ko XY ke aadhe se zyaada kholo. X se upar-neeche chaap, phir usi khulaav se Y se, jo A aur B par kaatein. Line AB, XY ko aadha 90° par kaat'ti hai. Kyun? AX = AY aur BX = BY, to △AXB ≅ △AYB (SSS). AB ka har bindu X aur Y se barabar door. Isse 90° angle bhi milta hai.

every point on AB: distance to X = distance to Y

Angle bisector: shirsh O se chaap kheencho jo bhujaon ko P aur Q par kaate. P aur Q se barabar chaap jo R par milein. Tab OR angle ko do barabar aadhon mein baant'ta hai (SSS se △OPR ≅ △OQR). 60° aur 90° ko baar-baar aadha karne se 30°, 15°, 45°, 22.5° …; prachin Śulba-sūtra vediyon ke liye aisi rachnaayein rassi aur khoonton se karte the.

60° → 30° → 15° · 90° → 45° → 22.5°

Angle ki nakal: puraane shirsh se dono bhujaon ko kaat'ta chaap, aur naye shirsh se waisa hi chaap. Bhujaon ke kaaton ka antar compass se naapo aur naye chaap par lagaao. Jodne par bilkul nakal, protractor ki zaroorat nahi. Transversal ke bindu P par angle copy karne se P se pehli line ke parallel line milti hai (barabar corresponding angles).

Hexagon, sitaare aur mehraab: compass ko vritt ke radius par rakhkar vritt par chalaao: ye theek 6 baar baithta hai, regular hexagon jiski bhuja radius ke barabar (chhah samabahu tribhuj, kendra par 60°). 60° angle seedhe samabahu tribhuj se milta hai (A se chaap, aur jahan wo line kaate wahan se waisa hi chaap), aur do milkar 120°, hexagon ka angle. Ek-chhod-ek kone jodne se 6 nok wala sitaara. Laal Qile ke Diwan-i-Aam jaisi sudaul mehraab ke liye pehle sahaayak lines: A aur D par barabar angles, aur AB = CD.

side of hexagon = radius

Explore: angle bisect karo.

  1. Angle bisect karne ke steps kram mein lagao.
  2. Ab A aur B se barabar door ek bindu lagaao.

Yaad rakho

  • Perpendicular bisector: dono siron se barabar chaap; iske bindu siron se barabar door.
  • Angle bisector, angle ki nakal, parallel aur hexagon sab barabar chaap aur sarvangsam tribhuj se.

Quick check

Radius apne hi vritt par kitni baar kadam rakhta hai?

  1. 6
  2. 4
  3. 8
  4. about 3.14
Answer

6. Har kadam kendra ke saath samabahu tribhuj: 6 × 60° = 360°.

Socho

Line l par na pade bindu P se l par lamb kaise kheenchoge?

Ek answer

P ko kendra lekar chaap kheencho jo l ko X aur Y par kaate. P, X aur Y se barabar door, to XY ke perpendicular bisector par hai. Wo bisector banao: P se guzarta aur l par lamb hai.

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.