Similar Triangles for 10th Indian CBSE

Similar TrianglesLesson 7: CBSE Text book, 2008S Chand& Co2 Ram’s Workshops PROPORTIONALITY THEOREMTheorem: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.GIVEN DE || BCTO PROVE:ECAEBDAD=3 Ram’s Workshops PROPORTIONALITY THEOREMCONSTRUCTIONAREA OF ΔADE=1/2 *AD*EF4 Ram’s Workshops PROPORTIONALITY THEOREM –Contd…AREA OF ΔBDE=1/2 *BD*EFAREA OF ΔADE=1/2 *AD*EFBDADEFBDEFADBDEareaADEarea==ΔΔ**21**215 Ram’s Workshops PROPORTIONALITY THEOREM –Contd…BDADBDEareaADEarea=ΔΔareaΔBDE=areaΔCDETriangles with the same basebetween two parallel linesSimilarly:CEAECDEareaADEarea=ΔΔCDEareaADEareaBDEareaADEareaΔΔ=ΔΔCEAEBDADTherefore=:But so far we have not used the other condition of DE || BC6 Ram’s Workshops Solve these Problemsa. In a triangle ABC, DE is drawn parallel to BC. Prove that AD*EC=AE*DBb. Prove that the line drawn from the midpoint Of one side of a triangle parallel to other side bisectsthe third side.7 Ram’s Workshops Angle Bisector TheoremABCEConstruction:Draw EB parallel to AD Which joins the extended AC at ED**ABCDABACTo=DBCD:prove **Proof:8 Ram’s Workshops Angle Bisector Theorem: proofEB and AD of lines parallel toal transversa is AB: see usLet :proved is theorem then theAB AE that provecan wei )(DBCDQED:Theoremality Proportion Basicper as HenceEB toparallel is AD:CEB=−−−−−−−=ΔfThusaAEACInQEDABAC−−−−−−−==Δ∠=∠∠=∠∠=∠∠=∠DBCDabove (a)in ngSubstitutiABAE Therefore triangleisoclesan is ABE thus--AEBABE(2) and (1) from Hencebisector angle theis AD as-given -DABCADBut -(2)---angles ingcorrespond ----AEBCAD :Thereforeal transversa also is AE and-(1)---angles alternate ---ABEDAB :thereforeABCDE9 Ram’s Workshops E7.2/9ACBEDcmECACAEECECButECECAEECAESidesbothonaddingECAETheoremiltyoporByDBADGiven8.138.435*88.4588.4ECAC8.4ACECAE 581......1.53..tanPr..53=−=−=======+=+=+==10 Ram’s Workshops E7.2/14ACBED1........'.....1 21...01.....012...0)1)(12(:..012...0224.......0224...072924927207218249151220)13)(78)35)(34(....35781334.......2222222==−==−=+=−+=−−=−−=++−=−+−+−+−−=+−−−−=−−−−=−−=xThereforenegativebetcanxxorxOrxorxxxgfactorizinxxorxxorxxOrxxxxxxxxxxxxxxtionmulitplicacrossbyxxxxeiECAEDBADGiven11 Ram’s Workshops E7.2/20In a trapezium the diagonals intersect in the same ratio3xnegative bet can' x 3 0 430)3)(34(0994018188244820104212245222=−==−+=−−=−−+−−=++−=−asrxxxxxxxxxxxxxxGiven12 Ram’s Workshops E7.2/25DCALDPDLandBLDCPLWeramParalleaisABCDGiven==......DP:prove tohave log......Construction:Draw a line parallel to BC and extend it to meet DC at K13 Ram’s Workshops E7.2/25…contdQEDDCALDPDLthereasALDKButDCDKDPDLSimilarly−−−−−−===ramparallelog a of sides theare ......;..QEDBLDCPLDPThereforeKCBLHenceRAMPARALLELOGAISKLBCbutKCDCPLDPtheoremalityproportionbyDCKCDPPLThereforeKLPCLDKtheGiven−−−−−====Δ:................:||.....14 Ram’s Workshops In fig given below DE||OQ and DF||OR. Show that EF||QRGiven: DE is || to OQDF is || to OR QRtoisEFThereomThalesperasthereforeFRPFEQPEHenceDOPDFRPFPORtheinAlsoDOPDEQPEPOQtheIn.....||..,.....................;....,.......==Δ=ΔD15 Ram’s Workshops Trapezium ProblemADCOBOCOBODAOTHATPROVECDABGIVEN=........||..:Simple: ΔAOB and ΔOCD are Similar –WHY?16 Ram’s Workshops SIMILAR TRIANGLES -RULESIf in two triangles: 1) corresponding angles are equal and 2) their corresponding sides are in the same ratiothen they are SIMILARAAAIf two angles of one triangle are respectively equalTo two angles of another triangle then the two trianglesAre SIMILARAAIf in two triangles: , sides of one triangle are proportionalTo the sides of other triangle, then their corresponding Angles are equal and hence the two triangles are SIMILARSSSIf one angle of a triangle is equal to another angle of a triangleAnd the sides including this angle are proportional then theTwo triangles are similarSASSIMILAR TRIANGLES HAVE THEIR SIDESOPPOSITE OF THEIR EQUAL ANGLES AREIN THE SAME RATIO17 Ram’s Workshops In the Fig the angles as shown are equal. Prove that PT*QR=PR*STST*PRQR*PT thatprove=Tox1122QEDPRSTQRPTOrQRSTTheseHENCEGIVENPQRPSTandangleQPRSPTInQPRSPTThereforexQPRxSPT−−−−−−−−−==∠=∠−−−−−∠=∠ΔΔ∠=∠+=∠+=∠..**..PRPT:ratio same in the are sides opposite The:thereforesimilarity-AA ---similar are s..triangle two........2..PST and PQR:Triangles Two following the:1118 Ram’s Workshops E7.3/24QEDPMBCPACAthereforeproportioninareTrianglesSimilarofAnglesofsidesOppositesSIMILARITYAAAMPABCTHEREFORECOMMONISAABCPMA−−−−−=−−ΔΔ∠∠=∠:..................:)..(~.......19 Ram’s Workshops E7.3/26PMADPQABThereforeSSSsimilararePQMandABDHenceangleIncludingtheisWhichcommonisBandQMBDPQABPQMABDandtheInQMBDQRBCQRBCPQABSOPMADPABoveToPQRABCGIVEN=−−ΔΔ∠=ΔΔ====ΔΔ:)(..............................;......2121.:Pr....~:20 Ram’s Workshops AREA OF SIMILAR TRIANGLES -1MBACPRQDThe ratio of the areas of two similar triangles is equalto the ratio of the squares of thethe corresponding sides)2(*21)1(*21ABC Area:are triangles theof are Re................&........&.....−−−−=−−−=ΔΔPMQRAreaPQRADBCHencespectivelyBasesvertexthefromdrawnlarsPerpendicuthearePMADsimilararePQRABCGiven)4(PQAB:similar are triangles twothe....)3(**21*21ABC Area−−−−−==−−−==PRACQRBCasAlsoPMADQRBCPMQRADBCAreaPQR21 Ram’s Workshops AREA OF SIMILAR TRIANGLES -2MBACPRQDOTHERSSIMILARLYPQABPQABAreaPQRingSusbstitutQRBCPQABPMADSofromQRBCButPMADPQABHenceAABIn..PQAB*ABC Area(3)on thesePQAB & )4(PQAB...... :ThereforeSIMILAR ARE PMQ. AND ADB ,)(90PMQADBalsosimilar are triangles theQ...as:PMQ and ADB : triangles220====−−−==ΔΔ−−=∠=∠∠=∠)4(PQAB:similar are triangles twothe....−−−−−==PRACQRBCasAlso22 Ram’s Workshops Some other proofs on areas of similar trianglesThe areas of two similar triangles are in the ratio of the square of the corresponding altitudes.The areas of two similar triangles are in the ratio of the squares of the corresponding medians.The areas of two similar triangles are in the ratio of the squares of the corresponding angle bisector segments.The areas of the two similar triangles are equal then the two triangles are congruent.23 Ram’s Workshops E7.5/64;25ABC of Area:ADE of 425ADE of AreaABC of Area251231sidesboth on one Adding ;23ADE of AreaABC of Area:ThereforeAAA) (similar are ADE and HenceDE and BC lines parallel twocuttung al transversa of ...ABC.andADEcommon isA :ADE and 22=ΔΔ==+==+=+=−−−−=ΔΔ∠=∠∠=∠∠ΔΔAreaOrADABADDBADADDBADDBTheoremADABABCanglesingCorrespondACBAEDABCInABC:ADE:determine To3:2DB:AD DE; ||BCiven ΔΔ=G24 Ram’s Workshops E7.5/22cmDNDNDNasDNAMSquareSquareAreaArea5.32744910049*25549100:esgiven valu thengSubstitutiTheorem -----similar are triangles two theAltitude theof Altitude theof DEF of ABC of 22222=======ΔΔ25 Ram’s Workshops E7.6/8CBADQEDBCBDOrBCABABBDhenceACGiven−−−==ΔΔ=∠=∠∠ΔΔ*AB:gmultiplyin cross :ThereforeAA)(similar are ABC and ABD 90DABACBcommon is BABC and ABDIn BDlar toperpendicuis20BADABC26 Ram’s Workshops Pythagoras and ConverseRead from Book 27 Ram’s Workshops An aero plane leaves an airport and flies north at a speed of 1000Kms/hr. at the same time another plane leaves west at a speed of 1200Kms/hr. after 1 ½hrs how far apart are they from each other?North???1000*1.5=1500KmsApplying Pythagoras theoremKms6130061*3*10061*3*3*100549*100225324*1001518*100)1500()1800(2222====+=+=+1200*1.5=1800KmsWest28 Ram’s Workshops E7.6/14()2222222222222222222222222222AB16CD efore4CD...TherBC )3(1622ABsidesboth on 2 .83AB(2) Rewriting4CDBCCD3CDBCCDBD know )2(ABabove (1)in AD of value theAD )1(ABThoerem Pythogoras From3CDBD BCACBCButCDACbygMultiplyinCDACABCDCDACWeBDCDACngsubstitutiCDACButBDADgiven+===−−−−+=+=+−===+=+−−−−+−=−=−−−−−+==29 Ram’s Workshops In an isosceles triangle ABC if AC=BC and AB2=2AC2, prove that ^C is a right angleQEDAngleRight theas C wtih Triangleangleright a of hypotenues theis AB ifonly possible is above thepythogorasper AC one of placein (1)in ngSubstitutigiven as esBC..IsosclAC ..)1(AB as written becan 2BA 22222222AsBCACABBUTACACwhichACGiven+==−−−−+==30 Ram’s Workshops L & M are the midpoints of AB and BC respectively of ΔABC, Right angled at B. Prove that 4LC2=AB2+4BC2ABCLQEDBCABTakingBCABingSusbstitutGivenABLBPythogorasBCLBIn22222222244LC4 of LCM 21LC21LCLBC +=+⎟⎠⎞⎜⎝⎛=−−−−=−−+=Δ31 Ram’s Workshops E7.7/212222AB:Prove ToBC lar toPerpendicu is AD DBACCDGiven+=+QEDBDACCDABarrangingBDCDACfrominCDADACADBinBDADAB222222222222222:ReAB(1)in (2) AD of value thengSubstituti(2)-----ADC )1( +=++−=Δ−−+=−−−Δ−−+=32 Ram’s Workshops Important InformationPlease note that the rhombus's diagonals cut each other at 900and equallyO33 Ram’s Workshops Interesting Sum 22DBAD:prove lyrespective larsPerpendicu andbisectors angle theare CD and CECat triangleangledright a is:EBAEToACBGiven=)1(EBAE:bisector angle theis CE:Pr2222−−−−−==BCACEBAEOrBCACAsoofContd….ADEBC34 Ram’s Workshops Contd…..ACABCD)2(*:CDBC:ThereforeruleAA ------similar are ACD and ACB Hence,commonA AnglesRight ---ADCACBACD and ACB2−−−−===ΔΔ∠∠=∠ΔΔADABACThereforeADACACABIn)3(*:similar are &2−−−===ΔΔDBABBCThereforeDBBCDCACBCABCDBABCSimilarlyContd….35 Ram’s Workshops Contd…..QEDDBADDBABADABBEAEngSubstitutiDBABBCADABACBCACBEAE−−−==−−−−=−−−−=−−−=**above (1)in )3(*)2(*)1(22222222Thanks for your attention