Invention of the plane geometrical formulae - Part II

Transcription

1 IOSR Journl of Mthemtics (IOSR-JM) e-issn: ,p-ISSN: X, Volume 6, Issue 3 (My. - Jun. 013), PP Invention of the plne geometricl formule - Prt II Mr. Stish M. Kple sst. Techer Mhtm Phule High School, Kherd Jlgon (Jmod) - 30 Dist- Buldn, Mhrshtr (Indi) bstrct: In this pper, I hve invented the formule for finding the re of n Isosceles tringle. My finding is bsed on pythgors theorem. I. Introduction mthemticin clled Heron invented the formul for finding the re of tringle, when ll the three sides re known. Similrly, when the bse nd the height re given, then we cn find out the re of tringle. When one ngle of tringle is right ngle, then we cn lso find out the re of right ngled tringle. Hence forth, We cn find out the re of n equilterl tringle by using the formul of n equilterl tringle. These some formule for finding the res of tringles re not exist only but including in eductionl curriculum lso. But, In eductionl curriculum. I don t ppered the formul for finding the re of n isosceles tringle with doing teching lerning process. Hence, I hve invented the new formul for finding the re of n isosceles tringle by using Pythgors theorem. I used pythgors theorem with geometricl figures nd lgebric equtions for the invention of the new formul of the re of n isosceles tringle. I Proved it by using geometricl formule & figures, 0 exmples nd 0 verifictions (proofs). Here myself is giving you the summry of the reserch of the plne geometricl formule- Prt II II. Method First tking n isosceles tringle BC B Fig. No. -1 C Now tking, & b for the lengths of three sides of BC B b C Fig. No. 10 Pge

2 Invention of the plne geometricl formule - Prt II Drw perpendiculr D on BC. B b/ D b/ C h b Fig. No. - 3 BC is n isosceles tringle nd it is n cute ngle lso. In BC, Let us represent the lengths of the sides of tringle with the letters,,b. Side B nd side C re congruent side. Third side BC is the bse. D is perpendiculr to BC. Hence, BC is the bse nd D is the height. Here, tking B= C = Bse, BC = b Height, D = h In BC, two congruent right ngled tringle re lso formed by the length of perpendiculr D drwn on the side BC from the vertex. By the length of perpendiculr D drwn on the side BC, Side BC is divided into two equl prts of segment. Therefore, these two equl segments re seg DB nd seg DC. Similrly, two right ngled tringles re lso formed, nmely, DB nd DC which re congruent. Thus, DB = DC = 1/ BC DB = DC = 1/ b = b/ DB nd DC re two congruent right ngled tringle. Tking first right ngled DC, In DC, Seg D nd Seg DC re both sides forming the right ngle. Seg C is the hypotenuse. Here, C = Height, D = h h DC = b/ nd m DC = 90 0 D b/ C Fig. No - ccording to Pythgors Theorem, (hypotenuse) = ( one side forming the right ngle) + ( second side forming the right ngle) In short, ( Hypotenuse ) = ( one side) + ( second side) C = D + DC D + DC = C h + ( b/ ) = h = (b/) h = b h = b h = b h = b 11 Pge

3 Invention of the plne geometricl formule - Prt II Tking the squre root on both side, h = b h = 1 ( b ) h = 1 - b The squre root of h is h nd the squre root of ¼ is ½.. h = ½ b.. Height, h = ½ b.. D =h = ½ b Thus, re of BC = ½ Bse Height = ½ BC D =½ b h But Height, h = ½ b.. re of BC = ½ b ½ b.. re of BC = b 1 b = b 1 b = b b.. re of n isosceles BC = b b 1 Pge

4 For exmple- Now consider the following exmples:- Ex. (1) If the sides of n isosceles tringle re 10 cm, 10 cm nd 16 cm. Find it s re Invention of the plne geometricl formule - Prt II D DEF is n isosceles tringle. In DEF given longside, 10cm 10 cm l ( DE) = 10 cm. l l ( DF) = 10 cm. Let, = 10 cm Bse, b = 16 cm. l ( EF) = 16 cm E 16 cm F Fig No- 5 By using The New Formul of n isosceles tringle,.. re of n isosceles DEF = ( DEF) = b - b = 16 (10) (16) = = = 1 The squre root of 1 is 1 = 1 = 8 sq.cm... re of n isosceles DEF = 8 sq.cm. Verifiction :- Here, l (DE) = = 10 cm. l ( EF) = b = 16 cm. l ( DF) = c = 10 cm. By using the formul of Heron s Perimeter of DEF = + b + c = = 36 cm Semiperimeter of DEF, S = + b + c S = 36 S = 18 cm...re of n isosceles DEF = s (s ) (s b) (s c) = 18 (18 10) (18 16) (18 10) = = (18 ) (8 8) = 36 6 = 36 6 The squre root of 36 is 6 nd the squre root of 6 is 8 = 6 8 = 8 sq.cm.. re of DEF = 8 sq.cm 13 Pge

5 Invention of the plne geometricl formule - Prt II Ex. () In GHI, l (GH) = 5 cm, l (HI) = 6 cm nd l (GI) = 5 cm. Find the re of GHI. n isosceles tringle. In GHI given longside, 5cm 5cm l ( GH ) = 5 cm. l ( HI ) = 6 cm. l ( GI ) = 5 cm H 6cm I Fig No- 6 G GHI is Let, = 5 cm Bse, b = 6 cm. By using The New Formul of re of n isosceles tringle,.. re of n isosceles GHI = b b = 6 (5) (6) The simplest form of 6 is 3 = 3 ( 5) 36 = = 3 6 The squre root of 6 is 8 = 3 8 = 3 8 = = 1 sq.cm... re of n isosceles GHI = 1 sq.cm. Verifiction :- Here, l (GH) = = 5 cm. l (HI) = b = 6 cm. l (GI) = c = 5 cm. By using the formul of Heron s Perimeter of GHI = + b + c = = 16 cm Semiperimeter of GHI, S = + b + c S = 16 S = 8 cm...re of n isosceles GHI = s (s ) (s b) (s c) 1 Pge

6 = 8 (8 5) (8 6) (8 5) = = (8 ) (3 3) = 16 9 = 1 Invention of the plne geometricl formule - Prt II The squre root of 1 is 1 = 1 sq.cm.. re of n isosceles GHI = 1 sq.cm. Explntion:- We observe the bove solved exmples nd their verifictions, it is seen tht the vlues of solved exmples by using the new formul of n isosceles tringle nd the vlues of their verifictions re equl. Hence, The new formul of the re of n isosceles tringle is proved. III. Conclusions:- re of n isosceles tringle = b b From the bove new formul, we cn find out the re of n isosceles tringle. This new formul is useful in eductionl curriculum, building nd bridge construction nd deprtment of lnd records. This new formul is lso useful to find the re of n isosceles tringulr plots of lnds, fields, frms, forests, etc. by drwing their mps. 1 Geometry concepts nd Pythgors theorem. References:- 15 Pge