Geometry1716.1 IntroductionGeometry is a branch of mathematics that deals with thepropertiesofvariousgeometricalfgures. Thegeometry whichtreatsthepropertiesandcharacteristicsofvarious geometricalshapeswithaxiomsortheorems,withoutthe helpofaccuratemeasurementsisknownastheoretical geometry.Thestudyofgeometryimprovesonespowerto think logically.Euclid, who lived around 300 BC is considered to be the father of geometry. Euclid initiated a new way of thinking inthestudyofgeometricalresultsbydeductivereasoning basedonpreviouslyprovedresultsandsomeselfevident specifc assumptions called axioms or postulates.Geometryholdsagreatdealofimportanceinfelds suchasengineeringandarchitecture.Forexample,many bridgesthatplayanimportantroleinourlivesmakeuse ofcongruentandsimilartriangles.Thesetriangleshelp toconstructthebridgemorestableandenablesthebridge towithstandgreatamountsofstressandstrain.Inthe constructionofbuildings,geometrycanplaytworoles; oneinmakingthestructuremorestableandtheotherin enhancing the beauty.Elegant use ofgeometric shapes can turn buildings and other structures such as the Taj Mahal into great landmarks admired by all. Geometric proofs play a vital role in the expansion and understanding of many branches of mathematics.The basic proportionality theorem is attributed to the famousGreekmathematicianThales.Thistheoremisalso calledThales theorem.GEOMETRYGEOMETRYEUCLID(300 BC) GreeceEuclids Elements is one ofthe mostinfuentialworksinthehistory of mathematics,servingasthemain textbookforteachingmathematics especially geometry.Euclids algorithm is an effcient methodforcomputingthegreatest common divisor. Introduction Basic ProportionalityTheorem Angle Bisector Theorem Similar Triangles Tangent chord theorem Pythagoras theoremThere is geometry in the humming ofthe strings, there is music in the spacing ofspheres - Pythagoras6617117210th Std. MathematicsA P1P2D P3B XECYTo understand the basic proportionality theorem, let us perform the following activity.Draw any angle XAY and mark points (say fve points), , , PPDP 1 2 3and B on arm AX such that1 AP P P P D DP P B 1 1 2 2 3 3 = = = = =unit (say).ThroughBdrawanylineintersectingarmAYatC. AgainthroughDdrawaline parallel to BC to intersect AC at E.NowAD=3 AP P P P D 1 1 2 2 + + =unitsandDB=2 DP P B 3 3 + =units` DBAD= 23Measure AE and EC.We observe that ECAE= 23Thus, inif ABC DE BC < D , then DBADECAE=We prove this result as a theorem known as Basic Proportionality Theorem or Thales Theorem as follows:6.2 Basic proportionality and Angle Bisector theoremsTheorem 6.1 Basic Proportionality theorem or Thales TheoremIfastraightlineisdrawnparalleltooneside ofatriangleintersectingtheothertwosides,thenit divides the two sides in the same ratio.Given: In a triangle ABC, a straight line l parallel toBC, intersects AB at D and ACat E.To prove:DBADECAE=Construction: Join BE, CD. DrawEF AB =andDG CA = .ProofSince,EF AB = , EF is the height of triangles ADE and DBE. Area ( ADE D )= 21# base # height =AD EF21# andArea( DBE D ) = 21# base # height =DB EF21#Fig. 6.1BAGECDFlFig. 6.2ActivityGeometry173` ( )( )DBEADEareaareaDD= DB EFAD EFDBAD2121##=(1)Similarly, we get ( )( )DCEADEareaareaDD= EC DGAE DGECAE2121# ## #=(2)But,DBE DandDCE Dare on the same base DE and between the same parallelstraightlinesBC and DE.`area ( ) DBE D = area ( ) DCE D(3)Form (1), (2) and (3), we obtain DBAD = ECAE. Hence the theorem.CorollaryIf in aABC D , a straight line DE parallel to BC, intersects AB at D and AC at E, then(i) ADABAEAC= (ii)DBABECAC=Proof (i) From Thales theorem, we have DBAD= ECAE( ADDB= AEEC (1 + ADDB= 1AEEC+( ADAD DB += AEAE EC +Thus, ADAB= AEAC(ii)Similarly, we can prove

DBAB= ECAC Is the converse of this theoremalso true? To examine this let us perform the following activity.Draw an angle+XAY and on theray AX, mark points, , , PPPP 1 2 3 4and B such thatAP P P P P P P P B 1 1 1 2 2 3 3 4 4 = = = = = unit (say).Similarly, on ray AY, mark points , , , QQ QQ 1 2 3 4and C such thatAQ QQ QQ QQ QC 2 1 1 2 2 3 3 4 4 = = = = =units (say).Ifbadc=then ba bdc d +=+.This is called componendo rule. Here, ADDBAEEC=& ADAD DBAEAE EC +=+by componendo rule.Do you know?Activity17410th Std. MathematicsNow joinP Q 1 1and BC.Then P BAP11= 41 and QCAQ11 = 82 = 41 Thus,P BAP11= QCAQ11We observe that the linesP Q 1 1and BC are parallel to each other.i.e., P Q BC 1 1 < (1)Similarly,by joining, P Q P Q 2 2 3 3andP Q 4 4we see that P BAPQCAQP Q BC32and22222 2 < = =(2) P BAPQCAQP Q BC23and33333 3 < = = (3) P BAPQCAQP Q BC14and44444 4 < = = (4)From (1), (2), (3) and (4) we observe that if a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.Inthisdirection,letusstateandproveatheoremwhichistheconverseofThales theorem.Theorem 6.2Converse of Basic Proportionality Theorem ( Converse of Thales Theorem)If a straight line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.Given: Aline lintersects the sides AB and ACofTABC respectively at D and Esuch that DBAD = ECAE(1)To prove :DE BC