Heron, Brahmagupta, Pythagoras, and the Law of Cosines

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1 University of Nersk - Linoln DigitlCommons@University of Nersk - Linoln MAT Exm Expository Ppers Mth in the Middle Institute Prtnership Heron, Brhmgupt, Pythgors, nd the Lw of Cosines Kristin K. Johnson University of Nersk-Linoln Follow this nd dditionl works t: Prt of the Siene nd Mthemtis Edution Commons Johnson, Kristin K., "Heron, Brhmgupt, Pythgors, nd the Lw of Cosines" (006). MAT Exm Expository Ppers This Artile is rought to you for free nd open ess y the Mth in the Middle Institute Prtnership t DigitlCommons@University of Nersk - Linoln. It hs een epted for inlusion in MAT Exm Expository Ppers y n uthorized dministrtor of DigitlCommons@University of Nersk - Linoln.

2 Heron, Brhmgupt, Pythgors, nd the Lw of Cosines Expository Pper Kristin K. Johnson In prtil fulfillment of the requirements for the Mster of Arts in Tehing with Speiliztion in the Tehing of Middle Level Mthemtis in the Deprtment of Mthemtis. Dvid Fowler, Advisor July 006

3 Johnson MAT Expository Pper - 1 Heron's Formul The formul for the re of tringle n e developed y mking n ext opy of the tringle nd rotting it 180. Then join it to the given tringle long one side to otin prllelogrm s shown ove. To form retngle, ut off smll tringle long the right nd join it t the other side of the prllelogrm. Beuse the re of the retngle is the produt of se () nd height (h), the re of the given tringle must e 1 h. The re of tringle is A = 1 h, where is the length of ny side of the tringle (the se) nd h (the ltitude) is the perpendiulr distne etween the se nd the vertex not on the se. This n e seen in the following sketh of ABC with se AB nd ltitude (height) CD. The height of tringle is not lwys given so this n limit the use of this formul to lulte the re of the tringle. Are formul = 1/h= C 1 = m ( ) AB CD CD = 3.67 m B D AB = 8.47 m A Now wht if you hve n inquisitive student who sks, "Wht if you know the sides of tringle ut don't know the height?" Enter Heron of Alexndri, Greek mthemtiin who lived in the first entury ( ). He ws elieved to hve tught t the Museum of Alexndri. This is due to the ft tht most of his writings pper s leture notes for ourses in mthemtis, mehnis, physis nd pneumtis. Heron's most fmous invention ws the first doumented stem engine. He lso wrote Metri ook tht desries how to find surfes nd volumes of vrious ojets. In this ook, there is proof for formul to find the re of ny tringle when the lengths of the sides re ll known lled Heron's Formul.

4 Johnson MAT Expository Pper - To use the formul, the semi-perimeter is lulted y finding the sum of the lengths of the sides of the tringle nd dividing the sum y. s = + + Then the re of the tringle is s(s )(s )(s ). An exmple using this formul to find the re is shown elow. = 4.48 m = 6.94 m = 8.47 m Semi perimeter s= ++ = 9.94 m Heron's formul A= s( s-) ( s-) ( s- ) = C = m ( ) ( ) ( ) Are BCA = m B Heron's proof of this formul ws found in Proposition 1.8 in his ook Metri. This doument ws lost until frgment ws found in 1894 nd full opy ws found lter in It is sid tht his proof is ingenious ut omplex. It is elieved though tht Arhimedes (87 B.C. - 1 B.C.) lredy knew the formul. A more essile proof, tht me from uses lger. Let A equl the re of tringle shown elow. Know tht = d + e nd h is n ltitude. Using the Pythgoren Theorem, proof of whih will e disussed in lter setion of this pper, d + h = nd h + e =. A Comining the system of equtions y sutrtion, d + h = h + e = ( ) h d + h ( e + h )= d e = d e ( d + e) ( d e)=

5 Johnson MAT Expository Pper - 3 Divide one side y d+e, whih is equivlent to dividing the other side y. ( d + e) ( d e) = d + e Add to oth sides of the eqution, then sustitute d+e in for nd simplify. d e + = + d e + d + e = + d = + d = + Now let's look t the re formul of the tringle with se nd height h. A = 1 h Using the ft h = d then h = ( d ) 1, I n sustitute in this expression for h. A = 1 ( d ) 1 ( ) ( d) ) 1 A = 1 Now sustitute in the vlue of d from ove nd egin simplifying the expression. A = 1 ( ) + A = 1 4( ) ( + ) ) 1 4A = ( ) ( + ) ) 1 16A = ( ) ( ) + By using the differene of squres, it n e ftored. 16A = ( + ( + )) ( ( + ) ( )( + + ) ( + + ) ( + ) 16A = + 1 The semi-perimeter of tringle is s = + + By using some lger eh of the following n e sustituted into the ove to get the next step simplified.

6 Johnson MAT Expository Pper - 4 s = + + nd ( s )= + + nd ( s )= + nd s 16A =16s s ( )( s ) ( s ) ( )( s ) ( s ) s( s ) ( s ) ( s ) A = s s A = Therefore, Heron's formul is proved. ( ) = + A student might sk then, "How would you find the re of other polygons? Does this formul work for qudrilterls?" Brhmgupt's Formul This question n e nswered y looking t Brhmgupt's Formul. Brhmgupt lived in Indi ( ). He wrote Brhm Sput Siddhnt of whih overed mny topis out stronomy nd mthemtis. He is ttriuted to the finding of zero. This is the erliest known text to tret zero s numer ll its own. It lso desries the use of positive numers s fortunes nd negtives s dets. In this text he desries formul for finding the re of yli qudrilterls whih is nmed fter Brhmgupt. It is n extension of Heron's Formul to four-sided polygon whose verties ll lie on the sme irle. Every tringle is yli, whih mens it n e insried in irle, nd tringle n e regrded s qudrilterl with one of its four edges set equl to zero. Brhmgupt's Formul sys tht the semi-perimeter of yli qudrilterl is s = d so the re of the yli qudrilterl is equl to ( s ) ( s ) ( s ) ( s d). The re of qudrilterl ABCD is equl to the sum of the res of ABD nd BCD. Are = 1 sin(a) + 1 d sin(c) Sine it s yli qudrilterl, DAB =180 DCBnd the sin(a) = sin(c). Then using this reltionship the re eomes: Are = 1 sin(a) + 1 d sin(a) Ftoring out ommon ftor of ½ sin(a) gives: Are = 1 sin(a) ( + d) Squre oth sides nd use the reltionship tht sin (A) = 1-os (A). Are = 1 4 sin (A)( + d) Are = 1 ( 4 1 os (A))( + d) Multiply oth sides of the eqution y 4 nd then use the distriutive property to rewrite the eqution.

7 Johnson MAT Expository Pper - 5 ( ) os (A)( + d) 4Are = + d Using the Lw of Cosines, whih will e proved lter in this pper, I n write the length of DB in two wys. + os(a) = + d d os(c) Sine A nd C re supplementry, then os (C) = -os(a). Sustitute this into the length of DB nd solve for os(a). + os(a) = + d + d os(a) + d = d os(a) + os(a) ( ) + d = os(a) d + + d ( ) d + = os( A) Use the vlue of the os(a) nd sustitute it into the re formul. Through ftoring it n e written s shown, 4Are = ( + d) + d ( + d) ( d + ) ( ) 16Are = 4( + d) + d 16Are = ( + + d ) ( + + d ) ( + + d ) ( + + d) The semi-perimeter is s = d, so s = d. This reltionship n e rewritten in the following wys: (s ) = + + d (s ) = + + d (s ) = + + d (s d) = + + d Now eh of the ove n e sustituted into the re formul nd then simplified in terms of the semi-perimeter. 16Are = (s )(s )(s )(s d) Are = (s )(s )(s )(s d) Are = (s )(s )(s )(s d) Therefore, Brhmgupt's Formul hs een proved. Brhmgupt's Formul n e extended to finding the re of other qudrilterls y dding nother term. A nd C re ny two pirs of opposite ngles. In yli qudrilterl eh pir of opposite ngles sum to π, so os(π/)=0 nd the finl term redues to 0. ( s ) ( s ) ( s ) ( s d) ( d) os A + C

8 Johnson MAT Expository Pper - 6 Heron's Formul is the speil se of Brhmgupt's Formul where the length d is equl to zero. This is similr to how the Pythgoren Theorem is speil se of the Lw of Cosines, whih will e disussed next. Pythgoren Theorem The Pythgoren Theorem sttes tht the sum of the squres of the legs of right tringle is equl to the squre of the hypotenuse. It n e used to find the length of side of right tringle if the other two sides re known. Pythgors lived round 560 B.C B.C. He ws Greek mthemtiin nd philosopher. He founded soiety sed on mysti, religious, nd sientifi ides. This soiety lived y strit ode of silene. It is sid Pythgoren would e put to deth if they disussed ides with people outside their irle. Whether Pythgors or someone else from his shool ws the first to disover proof of the theorem n't e determined. A sttement of the theorem ws disovered on Bylonin tlet round 1900 B.C B.C. Eulid's (. 300 B.C.) Elements furnished the first stndrd referene in Geometry nd he ws the first to prove tht the theorem ws "reversile". Hundreds of proofs hve een pulished, nd my possily hve more known proofs thn ny other theorem. United Sttes President Jmes Grfield devised n originl proof in Pythgoren Proposition y Elish Sott Loomis ontins over 350 proofs. An interesting note out the use of this formul in Hollywood is tht fter reeiving his rins from the wizrd in the 1939 film The Wizrd of Oz, the srerow inorretly sttes, "the sum of the squre roots of ny two sides of n isoseles tringle is equl to the squre root of the remining side." I found two proofs tht I prtiulrly like. The first is sed on the use of proportions. C B H A AC AB = AH BC nd AC AB = HB BC Using ross produts to rewrite the two proportions result in the following equtions. AC = AB AH nd BC = AB HB These n e omined using the ddition property of equlity nd then simplify. AC + BC = AB AH + AB HB ( ) AC + BC = AB AH + HB Notiing the ft tht AB = AH + HB, the eqution n e rewritten to AC + BC = AB AB or AC + BC = AB. Sine AC =, BC =, nd AB =, then + =.

9 Johnson MAT Expository Pper - 7 A visul proof, redited to Bhskr, my e more essile to my own students. Using right tringle, mke three opies nd rotte one 90, the seond 180, nd the third 70. Put them together without dditionl rottions so they form squre with side. Are of the lrge outer squre is whih is equl to the sum of the four tringles nd the smll inner squre. = ( ) + 4 = + + = + The Pythgoren theorem is used in more dvned mthemtis. The pplitions tht use the Pythgoren theorem inlude omputing the distne etween points on plne, onverting etween polr nd retngulr oordintes, omputing perimeters, surfe res nd volumes of geometri shpes. The following is n exmple tht might e used in n pplition prolem of the Pythgoren Theorem. A removl truk omes to pik up pole of length 6.5m. The dimensions of the truk re 3m, 3.5m, nd 4m. Will the pole fit in the truk? Students re enourged to drw digrm suh s the one shown elow to help them visulize the sitution. It ws generlized to tri-retngulr tetrhedron nd nother theorem lled de Gu's Theorem nd nother onnetion to the use of Heron's Formul to lulte the surfe re. De Gu's Theorem then the squre of the re of the fe opposite the right ngle orner is the sum of the squres of the res of the other three fes.

10 Johnson MAT Expository Pper - 8 The Pythgoren Theorem is limited to right tringles. So nother right student might sk, "Wht hppens to the formul if the we hd n ute or otuse tringle?" The Lw of Cosines The Lw of Cosines enters the piture. The Lw of Cosines reltes the osine of n ngle in tringle to its sides. For ny tringle with sides of lengths,,, nd with C the ngle opposite the side with length, = + os( C). The Lw of Cosines n e written for the other two sides in tringle: = + os( A) nd = + os( B). B h C i D -i A This lw generlizes to the Pythgoren Theorem, whih holds only for right tringles, so when C is equl to 90 its osine is 0, so = + os 90 ( ). Sine the os(90 ) = 0, then the term -(0) = 0 nd it simplifies to = +. It is useful for finding the length of the third side of tringle when two sides nd the inluded ngle re know. It lso llows you to lulte ngle mesures in tringle when the lengths of the three sides re given. This lw is proved looking t two ses, the first is when the tringle is ute nd the seond is when it is otuse. Cse 1- ute tringle using the figure ove: To prove the = + os(c), look t BDA. Using the Pythgoren Theorem = h + ( i). This n e rewritten through multiplition to get = h + i + i. In CBD, = h + i so h = i. This llows i sustituted in for h whih results in = ( i )+ i + i. By simplifying, ( ) to e = + i. In BCD, the trigonometri rtio for os(c) is i. Solving for i in the osine rtio leves i = os(c). Thus through sustitution, = + os(c) or = + os(c). Cse - otuse tringle using the figure elow: B h D i C A

11 Johnson MAT Expository Pper - 9 To prove the = + os(c), look t BDA. Using the Pythgoren Theorem, = h + + i ( ). This n e rewritten through multiplition ( ) to e to get = h + + i + i. In BCD, = h + i so h = i. This llows i sustituted in for h whih results in = ( i )+ + i + i. By simplifying, = + + i. In BCD, the trigonometri rtio for os(180 -C) is i. Solving for i in the osine rtio leves i = os(180 C) or i = os(c) Thus, through sustitution, = + + ( os(c)) or = + os(c). The onepts ehind the Lw of Cosines in the ook Elements y Eulid ( 300 B.C.) show up efore the generl use of the word "osine" y mthemtiins. Proposition 1 of Eulid's Elements desries the property for n otuse tringle nd Proposition 13 for n ute tringle. Al-Bttni ( 10 A.D.) generlized these results to spheril geometry. Al-Kshi ( A.D.) wrote the theorem in form suitle for tringultion. Its pplitions extend to lulting the neessry irrft heding to ounter wind veloity nd still proeed long desired ering to destintion. It hs prtil use in lnd surveying. Looking further into onnetions to the Lw of Cosines one would find out how it works for spheril tringles. Sine this is unit sphere, the lengths,, nd re equl to the ngles (in rdins) extended opposite n ngle y those sides from the enter of the sphere. For non-unit sphere, they re the distnes divided y the rdius. When looking t smll spheril tringles, tht is in whih the distnes,, nd re smll, the spheril lw of osines is out the sme s the lw of osines in plne. In spheril tringles oth sides nd ngles re usully treted y their ngle mesure sine sides re r lengths of gret irle. There is Lw of Cosines for the sides nd nother for the ngles. Using pitl letters to represent ngles, nd lower se to represent the opposite sides, the lw for sides is given s: os() = os()os() + sin()sin()os(a) os() = os()os() + sin()sin()os(b) os() = os()os() + sin()sin()os(c) nd the lw for ngles is given y: os(a) = os(b)os(c) + sin(b)sin(c)os() os(b) = os(a)os(c) + sin(a)sin(c)os() os(c) = os(a)os(b) + sin(a)sin(b)os()

12 Johnson MAT Expository Pper - 10 The Ntionl Counil of Tehers of Mthemtis disusses the need to mke onnetions etween mthemtil ides in the pulition Priniples nd Stndrds for Shool Mthemtis. When my students re enourged to think mthemtilly, they egin to look for nd mke onnetions inresing their mthemtil understnding. By reting these onnetions, they n uild new knowledge on previous understndings. The types of questions tht I need to sk my students re: Wht mde you think of tht? Why does this mke sense? How re these ides relted? Did nyone think out this in different wy? Students n develop new onnetions y listening to their lssmte's thinking s these nswers re disussed. New ides n then e seen s extensions of other mthemtil ides they lredy know. As hs een disussed in this pper, the ide tht one n generlize one mthemtil onept to new or more omplex prolem is very powerful nd often utilized tool for mthemtiins. It ws suggested to me to find nother pir of onepts tht er the sme reltionship s those I hve disussed. The following informtion ppers on the wesite ut-the-knot, y Andrew Bogomolny out generliztions. "Pirs of sttements in whih one is ler generliztion of nother wheres in ft the two re equivlent. 1. The Intermedite Vlue Theorem - The Lotion Priniple (Bolzno Theorem). Rolle's Theorem - The Men Vlue Theorem 3. Existene of tngent prllel to hord - existene of tngent prllel to the x-xis The Mlurin nd Tylor series. 6. Two properties of Gretest Common Divisor 7. Pythgors' Theorem nd the Cosine Rule 8. Pythgors' Theorem nd its prtiulr se of n isoseles right tringle 9. Pythgors' Theorem nd Lrry Hoehn's generliztion 10. Comining piees of nd N squres into single squre 11. Mesurement of insried nd (more generlly) sent ngles" I leve it to the reder to determine if these re relly proven.

13 Johnson MAT Expository Pper - 11 Referenes Ayou, A. B. (Spring 003) Why the Lw of Cosines on the sphere nd in the plne re equivlent. Mthemtis nd omputer edution. Bogomolny, A.( ). (001). Geometry to go. Wilmington, MA. Gret Soure Edution Group. Houghton Mifflin Compny. Ntionl Counil of Tehers of Mthemtis. (000). Priniples nd Stndrds for Shool Mthemtis. Reston, VA: Ppik, I. (005). Alger onnetions, onneting middle shool nd ollege mthemtis. Columi, MO. Prentie Hll. Serr, M. (1993). Disovering geometry, n indutive pproh. Berkeley, CA. Key Curriulum Press Computing_the_re_of tringle fter_d1 - explntion