Se ha denunciado esta presentación.
Se está descargando tu SlideShare. ×

Anuncio
Anuncio
Anuncio
Anuncio
Anuncio
Anuncio
Anuncio
Anuncio
Anuncio
Anuncio
Anuncio
Cargando en…3
×

1 de 7 Anuncio

# Proofs of Pythagorus Theorum.pptx

This proves the Great Pythagoras Theorem.

This proves the Great Pythagoras Theorem.

Anuncio
Anuncio

## Más Contenido Relacionado

Anuncio

### Proofs of Pythagorus Theorum.pptx

1. 1. Different Proofs of Pythagoras Theorem By Ansh,Prakul and Kushagra
2. 2. Proof 1 • ΔABF=ΔAECbySAS.Thisisbecause,AE= AB,AF=AC,and • ∠BAF=∠BAC+∠CAF=∠CAB+∠BAE= ∠CAE. • ΔABFhasbaseAFandthealtitudefromB equaltoAC.Itsareathereforeequalshalf thatofsquareonthesideAC.Ontheother hand,ΔAEChasAEandthealtitudefromC equaltoAM,whereMisthepointof intersectionofABwiththelineCLparallel toAE.Thus,theareaofΔAECequalshalf thatoftherectangleAELM.Whichsaysthat theareaAC²ofthesquareonsideAC equalstheareaoftherectangleAELM. • Similarly,theareaBC²ofthesquareon sideBCequalsthatofrectangleBMLD. Finally,thetwo rectanglesAELMandBMLD makeupthesquareonthehypotenuseAB.
3. 3. Proof 2 • Weknow,△ADB~△ABC • Therefore,AD/AB=AB/AC(correspondingsidesofsimilartriangles) • Or,AB² =AD×AC……………………………..……..(1) • Also,△BDC~△ABC • Therefore,CD/BC=BC/AC(correspondingsidesofsimilartriangles) • Or,BC² =CD×AC……………………………………..(2) • Addingtheequations(1)and(2)weget, • AB² +BC² =AD×AC+CD× AC • AB² +BC² =AC(AD+CD) • Since,AD+CD=AC • Therefore,AC² =AB² +BC² • Hence,thePythagoreantheoremisproved
4. 4. Proof 3 • Westartwithfourcopiesofthesametriangle.Threeof thesehavebeenrotated90°,180°,and270°, respectively.Eachhasareaab/2.Let'sputthem togetherwithoutadditionalrotationssothattheyforma squarewithsidec.Thesquarehasasquareholewiththe side(a-b).Summingupitsarea(a-b)² and2ab,the areaofthefourtriangles(4·ab/2),weget C² =(a-b)²+2ab =a²-2ab+b²+2ab =a²-b²
5. 5. Proof 4 • Two triangles are said to be similar if their corresponding angles are of equal measures and their corresponding sides are in the same ratio. Also, if the angles are of the same measure, then we can say by using the sine law, that the corresponding sides will also be in the same ratio. Hence, corresponding angles in similar triangles will lead us to equal ratios of side lengths.In triangle ABD and triangle ACB: • ∠A = ∠A (common) • ∠ADB = ∠ABC (both are right angles) • Thus, triangle ABD and triangle ACB are equiangular, which means that they are similar by AA similarity criterion. Similarly, we can prove triangle BCD similar to triangle ACB. Since triangles ABD and ACB are similar, we have AD/AB = AB/AC. Thus, we can say that AD × AC = AB2. Similarly, triangles BCD and ACB are similar. That gives us CD/BC = BC/AC. Thus, we can also say that CD × AC = BC2. Now, using both of these similarity equations, we can say that AC2 = AB2 + BC2. Hence Proved.
6. 6. Proof 5 • AlgebraicmethodproofofPythagorastheoremwillhelp usinderivingtheproofofthePythagorasTheoremby usingthevaluesofa,b,andc(valuesofthemeasuresof thesidelengthscorrespondingtosidesBC,AC,andAB respectively).ConsiderfourrighttrianglesABCwherebis thebase,aistheheightandcisthehypotenuse.Arrange thesefourcongruentrighttrianglesinthegivensquare, whosesideisa+b.Theareaofthesquaresoformedby arrangingthefourtrianglesisc2.Theareaofasquare withside(a+b)=Areaof4triangles+Areaofsquarewith sidec.Thisimplies(a+b)2 =4×1/2×(a×b)+c2,a2 +b2 +2ab= 2ab+c2.Therefore,a2 +b2 =c2.HenceProved.
7. 7. Proof 6 • Thisproofisavariationon#1,one oftheoriginalEuclid'sproofs.In parts1,2,and3,thetwosmall squaresareshearedtowardseach othersuchthatthetotalshaded arearemainsunchanged(and equaltoa²+b².)Inpart3,thelength oftheverticalportionofthe shadedarea'sborderisexactlyc becausethetwoleftovertriangles arecopiesoftheoriginalone.This meansonemayslidedownthe shadedareaasinpart4.Fromhere thePythagoreanTheoremfollows easily.