Page 9 - Bigyan Katha June 2021

mgvb bq)| ZvQvov (3, 4, 5) ‡_‡K hvÎv ïiæ K‡i ‡mLvbKvi mKj m`m¨‡K GKB msL¨v w`‡q ¸Y K‡i ‡hfv‡e bZzb Îqx‡`i ‡c‡q ‡Mjvg; g~j ÎqxwU A_©vr (3, 4, 5) ‡K ‡Zv ‡mfv‡e Ab¨ ‡Kv‡bv Îqx ‡_‡K ‰Zwi K‡i ‡bIqv hvq wb| ¸Y wKsev fv‡Mi g‡a¨ w`‡q Zv‡K Lyu‡R Avbv hvq wb| ZvB (30, 40, 50) ev (21, 28, 35) wbQKB mvaviY wc_v‡Mvixq Îqx; ‡gŠwjK Îqx bq| G‡`i A‡gŠwjK wc_v‡Mvixq Îqx ejv ‡h‡Z cv‡i|
Z‡eGBwÎfRz ¸wjicwÖZwUBn‡eGgb GKwUmg‡KvYxwÎfRz ,hv(3,4,5)wÎf‡zRi m`k„ | m`k„ wÎfRz (Similar triangles) ej‡Z Kx ‡evSvq Zv m¤¢eZ mK‡jiB g‡b Av‡Q, Zey Zv GKevi ¯§iY Kwi‡q ‡`Iqv hvK| hLb `wy U wÎf‡z Ri ‡KvY¸wj ci¯úi mgvb nq, wKš‘ evû¸wj ci¯úi Amgvb _v‡K‡miKg`wyUwÎfRz ‡Kejvnqm`k„ wÎfRz | G ‡hb KZKUv ‡QvU-eo `By ‡ev‡bi gZ hv‡`i ‡`L‡Z GKB iKg wKš‘ eq‡mi Zdvr _vKvq Zviv AvKv‡i ‡QvU-eo| m`k„ , wKš‘ mem© g bq| ZvB (3, 4, 5) Avi (30, 40,50)ev(21,28,35)wÎfRz ¸wjevG‡`i ‡h ‡Kv‡bvwUi cvk¸©¦ wj‡K GKB msL¨v w`‡q ¸YK‡icvIqvmemg‡KvYxwÎfRz ¸wjn‡e ci¯ú‡ii m`k„ | Avgiv ‡`L‡Z cv‡ev GKwU ‡gŠwjK wc_v‡Mvixq Îqx w`‡q MwVZ mg‡KvYx wÎf‡z Ri AmsL¨ m`k„ wÎfRz cvIqv m¤¢e; ‡h wÎf‡z Ri cvk¸©¦ wj c‡Ö Z¨‡KB Avgv‡`i ‡`‡e GK GKwU mvaviY wc_v‡Mvixq Îqx| wKš‘ ‡Kv‡bvbZbz ‡gŠwjKwc_v‡MvixqÎqx‡mLv‡b cvIqv hv‡e bv| Ab¨fv‡e ej‡j, GKwU ‡gŠwjK wc_v‡Mvixq Îqx w`‡q ‡h mg‡KvYx wÎfRz MwVZ nq Zv KLbB Avi GKwU ‡gŠwjK wc_v‡Mvixq Îqx w`‡q MwVZ wÎf‡z Ri m`k„ n‡e bv| Zvi KviY ‡mB GKB| GKwU ‡gŠwjK Îqxi m`m¨‡`i ‡KvbiK‡gi msL¨v w`‡q ¸Y ev fvM K‡i KLbB Avi GKwU ‡gŠwjK wc_v‡Mvixq Îqx cvIqv hvq bv|
cwÖ ZwU ‡gŠwjK wc_v‡Mvixq Îqx GB A‡_©
Abb¨| GB Av‡jvPbvq c‡i Avgv‡`i Avevi Avm‡Z n‡e|
jÿKi‡j‡`Lvhv‡eZjz bvq‡QvUmsL¨v w`‡q MwVZ ‡h me ‡gŠwjK wc_v‡Mvixq Îqxi m‡1⁄2 Avgv‡`i cwiPq LvwbKUv ‡ewk, Zviv wKš‘ mK‡jB ‡Rvo I we‡Rvo msL¨vi mswgk‡Ö Y MwVZ| hw` ZvB nq, Zvn‡j ‡mB ‡Rvo- we‡Rv‡oi wbev© Pb Ki‡Z n‡e Kxfv‡e? wVK
Kx wbq‡g ‡g‡b Pj‡j Zv‡`i nw`k cvIqv hv‡e? ‡mB wePv‡i ‡XvKvi Av‡M bx‡Pi mviwY ‡_‡K welqwU cvÖ _wgKfv‡e e‡y S ‡bIqv hvK; ‡`Lv hvK ‡gv‡Ui Ici ‡Rvo-we‡Rvo msL¨vi wVK ‡Kvb ai‡Yi mswgkYÖ Avgv‡`i ‡gŠwjK wc_v‡Mvixq Îqx w`‡Z cv‡i| Avevi KLb ‡Rvo- we‡Rvo Dcw¯Z’ ‡_‡KI Avgv‡`i wbQK mvaviY Îqx Dcnvi ‡`q Ges ‡KbB ev Zv ‡`q|
bx‡PimviwY‡ZIb¤i^ mvwi‡Z‡`Lv
hv‡”Q ‡h, GB wbev© Pb m¤¢e bq, hvi KviY Avgiv Av‡MB e‡jwQ| GB ‡ÿ‡Î wc_v‡Mvixq m¤ú‡Ki© mgxKiY wP‡ýi GKcv‡k ‡`Lv ‡`‡e ‡Rvo msL¨v (a2 + b2) Avi Aci cv‡k ‡`Lv hv‡e we‡Rvo msL¨v (c2)| mviwY‡Z II b¤i^ ‡ÿ‡Î ‡`Lv hv‡”Q ‡h, a, b, c G‡`i cwÖ ZwU ‡Rvo msL¨v n‡j wc_v‡Mvixq Îqx MVb Kiv m¤¢e Z‡e Zv ‡gŠwjK wc_v‡Mvixq Îqx n‡e bv| GB Av‡jvPbvI Av‡M Kiv n‡q‡Q| IV
I V cKÖ Z... c‡ÿ GKB NUbv weeZ„ Ki‡Q| ‡gŠwjK wc_v‡Mvixq Îqx MV‡bi ‡ÿ‡Î mg‡KvYx wÎf‡z Ri `wy U evûi GKwUi (a A_ev b) nIqv PvB ‡Rvo Ges AciwU‡K n‡Z n‡e we‡Rvo| Z‡e GB RvZxq ‡Rvo-we‡Rv‡oi mgšq^ mvaviY Îqxi ‡ÿ‡ÎI NU‡Z cv‡i| GB welqwU wb‡q AviI K_v ejvi Av‡M Avgiv mviwYi III b¤‡^ i ewYZ© m¤¢vebvi ‡ÿÎwU
wb‡q Av‡jvPbv Ki‡ev| ‡Kb GB m¤¢vebv AvcvZ`w„ó‡ZwVKg‡bn‡jIMnÖYKivhv‡”Q bv ‡`Lv hvK|
mviwYi μwgK msL¨v III Gi ‡ÿ‡Î ‡`Lv hv‡”Q ‡h (a2+ b2) Ges c2 GB `wy UB ‡Rvo msL¨v Zvn‡j ‡Kb a, b Ges c i GB wbev© Pb
Ryb 2021 9 ‡g‡b ‡bIqv hv‡e bv? jÿ Ki‡j ‡`Lv hv‡e
GLv‡b a Ges b Df‡qB we‡Rvo msL¨v, Zvn‡j Avgiv GB `wy U ivwk‡K mvaviYfv‡e wjL‡Z cvwi, a = 2m +1, b = 2n +1, ‡hLv‡b m Ges n `ywU ALÛ abvZ¥K msL¨v| ‡m‡ÿ‡Î Avgiv cv‡ev,
a2 + b2 = (2m +1)2 + (2n +1)2
= 4(m2 + n2) + 4(m + n) + 2
= c2 = ‡Rvo msL¨v; c2 hLb GKwU eM© ivwk Ges ‡Rvo msL¨v Zv Aek¨B 4 Øviv wefvR¨ n‡e; KviY GB c2 GK eM© ivwk wn‡m‡e Av`‡Z `ywU mgvb ‡Rvo msL¨vi ¸Ydj| ZvB bv?
wKš‘ Avgiv (a2 + b2) Gi gv‡bi ‡h e¨Äbv cvw”Q ‡mw`‡K hw` bRi w`B A_©vr [4(m2 + n2 ) + 4(m + n) + 2] ‡K jÿ Kwi Zvn‡j ‡`L‡Z cv‡ev ‡h GB e¨Äbvi cÖ_g `ywU c` 4 Øviv wefvR¨ n‡jI ‡kl c` 2 wKš‘ wefvR¨ bq| AZGe a Ges b Df‡qB we‡Rvo msL¨v n‡j Zv‡`i e‡M©i ‡hvMdj ‡Rvo msL¨v n‡e wVKB wKš‘ Zv ‡Kv‡bv c~Y©eM© msL¨v ‡`‡e bv| ZvB Avgiv hw` mg‡KvYx wÎfz‡Ri `ywU evû‡K we‡Rvo msL¨v wn‡m‡e wbB Zvn‡j wKš‘ AwZfzR wn‡m‡e ‡Kv‡bv ALÛ ivwk cv‡ev bv, A_©vr ‡gŠwjK ev mvaviY, ‡Kv‡bv ai‡YiB, wc_v‡Mvixq Îqx MVb Kiv m¤¢e n‡e bv| MvwYwZK `„wó‡KvY ‡_‡K ejv hvq ‡h `ywU we‡Rvo msL¨vi e‡M©i ‡hvMdj Av‡iKwU c~Y©eM© ivwk w`‡Z cv‡i bv| GKUv mvaviY Dcvq Aej¤^b K‡i GUv Ic‡i ‡`Lv‡bv n‡q‡Q| wKQy wbw`©ó msL¨v wb‡q D`vniY w`‡jI GKB welq ‡`Lv hv‡e|
GLvb ‡_‡K ‡h wm×v‡šÍ Avgiv Avm‡Z cvwi Zv nj, mviwY ÔGKÕ Gi ‡Kej wZbwU (II, IV, Ges V) ‡ÿÎ ‡_‡K m¤¢ve¨ wbev© Pb ‡_‡K wc_v‡Mvixq Îqx cvIqv m¤¢e|
Zvi g‡a¨ μwgK msL¨v II ‡_‡K ‡gŠwjK wc_v‡Mvixq Îqx cvIqv hv‡e bv, Zv wgj‡Z cv‡i ‡Kej IV wKsev V G ewYZ© c‡_| AveviIVIVe¯ÍZy t`wyUGKBK_vej‡Q,
mviwY GKt (wc_v‡Mvixq Îqx MV‡bi Rb¨ a, b Ges c Gi ‡ÿ‡Î ‡Rvo I we‡Rvo msL¨vi m¤¢ve¨ web¨vm)
μwgK msL¨v
evû a
a2
evû b
b2
a2 + b2
AwZfyR c
c2
gšÍe¨
I
we‡Rvo
we‡Rvo
we‡Rvo
we‡Rvo
‡Rvo
we‡Rvo
we‡Rvo
‡Kv‡bviKg Îqx m¤¢e bq
II
‡Rvo
‡Rvo
‡Rvo
‡Rvo
‡Rvo
‡Rvo
‡Rvo
m¤¢e, ‡gŠwjK Îqx n‡e bv
III
we‡Rvo
we‡Rvo
we‡Rvo
we‡Rvo
‡Rvo
‡Rvo
‡Rvo
AvcvZ`„wó‡Z m¤¢e g‡b n‡jI cÖK...Zc‡ÿ m¤¢e bq; KviY Dc‡i Av‡jvwPZ n‡q‡Q
IV
we‡Rvo
we‡Rvo
‡Rvo
‡Rvo
we‡Rvo
we‡Rvo
we‡Rvo
‡gŠwjK I mvaviY Îqx cvIqv ‡h‡Z cv‡i
V
‡Rvo
‡Rvo
we‡Rvo
we‡Rvo
we‡Rvo
we‡Rvo
we‡Rvo
‡gŠwjK I mvaviY Îqx cvIqv ‡h‡Z cv‡i