MATHEMATICS STUDY MATERIAL PROPERTIES AND SOLUTIONS OF TRIANGLES & HEIGHTS AND DISTANCES AIEEE NARAYANA INSTITUTE OF CORRESPONDENCE COURSES

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1 MTHEMTIS STUDY MTERIL PROPERTIES ND SOLUTIONS OF TRINGLES & HEIGHTS ND DISTNES IEEE NRYN FNS HOUSE, 6 KLU SRI MRKET SRVPRIY VIHR, NEW DELHI-006 PH.: (0) 00//50 FX : (0) 880 Wesite : w w w. n r y n i. m E-mil : i n n r y n i. m

2 00 NRYN GROUP This study mteril is prt f NRYN fr IEEE, This is ment fr the persnl use f thse students wh re enrlled with NRYN, FNS Huse, 6, Klu Sri Mrket, New Delhi-006, Ph.: 00//50. ll rights t the ntents f the Pkge rest with NRYN INSTITUTE. N ther Institute r individul is uthrized t reprdue, trnslte r distriute this mteril in ny frm, withut prir infrmtin nd written permissin f the institute.

3 PREFE Der Student, Hertiest ngrtultins n mking up yur mind nd deiding t e n engineer t serve the siety. s yu re plnning t tke vrius Engineering Entrne Exmintins, we re sure tht this STUDY PKGE is ging t e f immense help t yu. t NRYN we hve tken speil re t design this pkge rding t the Ltest Pttern f IEEE, whih will nt nly help ut ls guide yu t mpete fr IEEE & ther Stte Level Engineering Entrne Exmintins. The slient fetures f this pkge inlude :! Pwer pked divisin f units nd hpters in sientifi wy, with rreltin eing there.! Suffiient numer f slved exmples in Physis, hemistry & Mthemtis in ll the hpters t mtivte the students ttempt ll the questins.! ll the hpters re fllwed y vrius types f exerises (Level-I, Level-II, Level-III nd Questins sked in IEEE nd ther Engineering Exms). These exerises re fllwed y nswers in the lst setin f the hpter. This pkge will help yu t knw wht t study, hw t study, time mngement, yur weknesses nd imprve yur perfrmne. We, t NRYN, strngly elieve tht qulity f ur pkge is suh tht the students wh re nt frtunte enugh t ttend t ur Regulr lssrm Prgrms, n still get the est f ur qulity thrugh these pkges. We feel tht there is lwys spe fr imprvement. We wuld welme yur suggestins & feedk. Wish yu suess in yur future endevurs. THE NRYN TEM KNOWLEDGEMENT While prepring the study pkge, it hs eme wnderful feeling fr the NRYN TEM t get the whleherted supprt f ur Stff Memers inluding ur Designers. They hve mde ur j relly esy thrugh their untiring effrts nd nstnt help t every stge. We re thnkful t ll f them. THE NRYN TEM

4 ONTENTS O N T E N T S Thery PROPERTIES ND SOLUTIONS OF TRINGLES ND HEIGHTS & DISTNES Slved Exmples Exerises Level I Level II Level III Questins sked in IEEE nd ther Engineering Exms

5 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN PROPERTIES ND SOLUTIONS OF TRINGLES & HEIGHTS ND DISTNES IEEE Syllus Prperties f tringles inluding entrid, inentre, irumentre nd rthentre, slutin f tringles, Heights nd distnes. ONTENTS Prperties f Tringle Elements f tringle Sine Rule sine Rule Prjetin frmule Tngent Rule Hlf ngle frmule m-n Therem re f Tringle entrid nd Medins f Tringle isetrs f the ngles irum irle Inirle Esried irles Slutin f Tringles Orthentre nd Pedl tringle f tringle Exentrl Tringle Distnes etween the speil pints Results relted with plygns Heights nd Distnes INTRODUTION tringle hs three ngles nd three sides. There re mny reltins mng these six quntities whih help in studying the vrius prperties f tringle. Fr exmple, if ny three quntities ut f three ngles nd three sides (t lest ne f whih is side), re given then using sme f these reltins, the remining three n e fund, whih is termed s the slutin f the tringle. FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

6 NRYN Mthemtis : Slutin f Tringle & Heights nd Distnes PROPERTIES OF TRINGLE. ELEMENTS OF TRINGLE In tringle, the ngles re dented y pitl letters, nd nd the length f the sides ppsite t these ngles re dented y smll letters, nd. Semi perimeter f the tringle is + + given y s nd its re is dented y. Nte : (i) + + (ii) + >, + >, + >. SINE RULE In tringle, the sides re prprtinl t the sines f the ngles ppsite t them i.e. sin sin sin Illustrtin : In ny tringle, prve tht s + s + s sin sin Slutin : Let k, Then, sin, k sin, k sin sin sin sin L.H.S. s + s + s k sin s + k sin s + k sin s. OSINE RULE k [sin + sin + sin] k (sin sin sin) k sin sin sin sin sin R.H.S. [ k sin ] + () s + () s r +. s r +. s () + s r +. s Illustrtin : If the lengths f the sides f tringle re x + x +, x, nd x + then find the gretest ngle? Slutin : Given the sides x + x +, x, x + putting x, we get the vlues f sides,, s, 0, respetively Similrly putting x, we get 7,, 5 respetively Gretest side ppsite ngle is gretest ngle s ()(5) (5) 0 FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

7 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN. PROJETION FORMULE In ny, () s + s () s + s () s + s Illustrtin : If 5 0, 75 0, prve tht +. Slutin : s 5 0, 75 0 we hve 60 0 R.H.S. ( s + s ) ( s s 5 0 ) + L.H.S. 5. NPIER S NLOGY (TNGENT RULE) In ny, () tn + t () tn + t () tn + t Illustrtin : In, if ( ) x tn tn, then find x + y + z (in terms f x, y, z)? ( ) y tn tn, ( ) z tn tn Slutin : x, y, z then put,, then x, 5 y, z x + y + z xyz 0 6. HLF NGLE FORMULE In ny, () (i) sin (iii) sin ( s )( s ) ( )( ) s s (ii) sin ( s )( s ) FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

8 NRYN s( s ) () (i) s ( ) ss (iii) s ( )( ) () (i) tn ( ) s s ss s(s ) Mthemtis : Slutin f Tringle & Heights nd Distnes s( s ) (ii) s ( s )( s ) (ii) tn ( ) ss s(s ) (iii) tn ( s )( s ) ( ) s s s(s ) + +, where s nd re f tringle. Nte : s(s )(s )(s ) () (i) sin ss ( )( s )( s ) (ii) sin ss ( )( s )( s ), (iii) sin ss ( )( s )( s ) Illustrtin 5: In if t/ t/, t/ t/ nd t/ t/ then + +? s s s Slutin : t/ t/ s(s ) s(s ) (s )(s ) (s )(s ) s s s s Similrly, s s s s + + s + + s s s s s 7. m-n THEOREM Let D e pint n the side f suh tht D : D m : n nd D θ, D α nd D β (s shwn in figure). Then () (m + n) t θ m t α n t β FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

9 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN () (m + n) t θ n t m t α β θ m D n 8. RE OF TRINGLE The re f tringle is given y sin sin sin s(s )(s )(s ) (Her s frmul) R Illustrtin 6: If in tringle, 6, nd s ( ) 5, find the re f the tringle. Slutin : 6, s( ) tn 5 tn t + 6 t 6 + re f 6 9 sq.units 9. ENTROID ND MEDINS OF TRINGLE The line jining ny vertex f tringle t the mid pint f the ppsite side f the tringle is lled the medin f the tringle. The three medins f tringle re nurrent nd the pint f nurreny f the medins f ny tringle is lled the entrid f the tringle. The entrid divides the medin in the rti :. The lengths f the medins thrugh, nd respetively re given y θ m D n α β m +, m + nd m + Nte : m + m + m ( + + ) FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880 5

10 NRYN Mthemtis : Slutin f Tringle & Heights nd Distnes 0. ISETORS OF THE NGLES If D isets the ngle nd divide the se int prtins x nd y, we hve, y Gemetry, x y x + y x y + + x + nd y + ls let δ e the length f D we hve D + D δ sin + δ sin sin, x D δ y sin i.e., D δ s + + sin Illustrtin 7: If the isetr f ngle f the tringle mkes n ngle θ with, then sinθ? Slutin : We hve D +, D + sinθ sin / Frm tringle D, + + sin + sin sinθ sin / sin / sin D ( + ) ( ) sin s ( ) sin / s sin/s/. IRUM IRLE The irle whih psses thrugh the ngulr pints f, is lled its irumirle. The entre f this irle i.e., the pint f nurreny f the perpendiulr isetrs f the sides f the, is lled the irumentre. Rdius (R) f the irumirle is given y the fllwing frmule R sin sin sin Illustrtin 8 : 6 / D / If the distnes f the sides f frm its irumentre e x, y nd z respetively, then prve tht + + x y z xyz. Slutin : Let M e the irumentre. MD. S D D nd MD. In DM, D tn r MD tn, i.e., tn, x x FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880 E O F

11 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN Similrly, y tn, tn z tn + tn + tn + + x y z F z M E y nd tn. tn. tn.. x y z ut in tringle, tn + tn + tn tn. tn. tn + + x y z xyz. x D. INIRLE The irle whih n e insried within the tringle s s t tuh eh f the sides f the tringle is lled its inirle. The entre f this irle i.e., the pint f nurreny f ngle isetrs f the tringle is lled the inentre f the. Rdius f the Inirle is given y the fllwing frmule r s (s ) tn (s ) tn (s ) tn R sin sin sin sin sin sin sin sin sin s s s E r I 0 90 / / r D r / F Illustrtin 9 : Find the distne etween the irumentre nd the inentre. Slutin : Let O e the irumentre nd I e the inentre f. Let OF e perpendiulr t nd IE e perpendiulr t. 0 OF 90. OI IF OF + + ( 0 90 ) + IE r ls, I R sin sin sin sin F O I E OI O + I O. Is OI R + 6R sin sin 8R sin sin s OI + 6sin sin 8sin sin s s + sin sin R 8 sin sin s s sin sin 8 sin sin sin...(i) OI R 8sin sin sin. 7 FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

12 NRYN Mthemtis : Slutin f Tringle & Heights nd Distnes. ESRIED IRLES The irle whih tuhes the side nd the tw sides nd prdued is lled the esried irle ppsite the ngle. Its entre nd rdius will e dented y I nd r respetively. Rdii f the exirles re given y the fllwing frmule () r s s stn Rsin s s s s F D E () r s s stn Rs sin s s s L I M () r s s stn Rs s sin s s Illustrtin 0: If d, d, d re the dimeters f the three esried irles f tringle, then d d + d d + d d? Slutin : d d + d d + d d (r r + r r + r r ) s (s) ( + + ) 8 FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

13 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN SOLUTION OF TRINGLES When ny three f the six elements (exept ll the three ngles) f tringle re given, the tringle is knwn mpletely. This press is lled the slutin f tringles. + (i) If the sides, nd re given, then s. nd n e tined in the similr wy. (ii) If tw sides nd nd the inluded ngle re given, then using tn + t, we get. + ls 90 0, s tht nd n e evluted. The third side is given y sin sin. (iii) If tw sides nd nd the ngle (ppsite t side ) re given, then sin sin, 80 0 ( + ) nd sin give the remining elements. If < sin, there is n tringle sin pssile (fig ). If sin nd is n ute ngle, then there is nly ne tringle pssile (fig ). If sin < < nd is n ute ngle, then there re tw vlues f ngle (fig ). If < nd is n ute ngle, then there is nly ne tringle (fig ). sin sin (Fig ) D (Fig ) D sin sin (Fig ) D (Fig ) This se is, smetimes, lled n miguus se. FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880 9

14 NRYN Mthemtis : Slutin f Tringle & Heights nd Distnes Illustrtin : In ny tringle, the sides re 6 m, 0 m nd m. Shw tht the tringle is tuse-ngled with the tuse ngle equl t 0 0. Slutin : Let, 0, 6 The lrgest ngle is ppsite the lrgest side. s ORTHOENTRE ND PEDL TRINGLE OF TRINGLE In tringle the ltitudes drwn frm the three verties t the ppsite sides re nurrent nd the pint f unurreny f the ltitudes f the tringle is lled the rthentre f the tringle. The tringle frmed y jining the feet f these perpendiulrs is lled the pedl tringle i.e. DEF is the pedl tringle f. F E P 0 90 D The tringle DEF whih is frmed y jining the feet f the ltitudes is lled the pedl tringle. () The distnes f the rthentre frm the ngulr pints f the re R s, R s nd R s () The distnes f P frm sides re R s s, R s s nd R s s () The sides f the pedl tringle re s ( R sin ), s ( R sin ) nd s ( R sin ) nd its ngles re, nd. () irumrdii f the tringles P, P, P nd re ll equl. Illustrtin : Find the distne f the rthentre frm the sides nd ngulr pints f tringle. Slutin : PD D tn PD D tn (90 0 ) s t sin s s R s s Similrly, PE R s s nd PF R s s gin, P E se D s se S, sin s R s P R s nd P R s F 90 P D E 0 FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

15 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN 5. EXENTRL TRINGLE The tringle frmed y jining the three exentres I, I nd I f is lled the exentrl r exentri tringle. Nte tht : () Inentre I f is the rthenter f the exentrl I I I. () is the pedl tringle f the I I I. () the sides f the exentrl tringle re R s, R s nd R s nd its ngles re, nd. () II R sin : II R sin : II R sin 6. THE DISTNES ETWEEN THE SPEIL POINTS () The distne etween irumenter nd rthenter is R. 8s ss () The distne etween irumenter nd inentre is R Rr. () The distne etween inentre nd rthenter is r R s ss. 7. RESULTS RELTED WITH POLYGONS Sme gemetril results relted with plygns re s fllws: () In regulr plygn, ll sides re equl. () Numer f verties f plygn numer f sides f the plygn. i.e. if plygn is f n sides, numer f its verties will e n. () Let... n e regulr plygn f n sides nd let O e the entre f this plygn. lerly, this O will ls e the entre f the irumsriing irle t this plygn. n O θ Nw, O O O... no θ (suppse). FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

16 NRYN Mthemtis : Slutin f Tringle & Heights nd Distnes Then, n θ θ n. i.e., ngle sutended y eh side f regulr plygn f n sides t its entre () Sum f interir ngles f the plygn + n sum f Interir ngles f the plygn f n sides n. (n ) (n ) (5) Perimeter (P) nd re () f regulr plygn f n sides insried in irle f rdius r re given y P nrsin n nr sin n Perimeter nd re f regulr plygen f n sides irumsried ut given irle f rdius r is given y P nrtn n nr tn n (6) If,,, d e the sides nd s the semiperimeter f yli qudrilterl, then its re (s )(s )(s )(s d) Nte : If we hve ny qudrilterl, nt neessrily insrile in irle, we n express its re in terms f its sides nd the sum f ny tw ppsite ngles. d D Fr let the sum f the tw ngles nd D e dented y α, then the re f the qudrilterl (s )(s )(s )(s d) d s α 8. TIPS ND TRIKS () In n equilterl tringle nd 60 (i) + + S (ii) s(s )(s )(s ) FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

17 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN (iii) r (iv) s.. R. (v) r s / r r r h p p p ( p, p, p re the lengths f the ttitudes f tringle) (vi) r : R : r : : () In right ngled tringle, right ngled t, nd 5,, then (i) 5+ + s 6 (ii) (iii) r s (iv) ()() 6 R (5)()() 5 (6) 6 (v) r 6 s 6 (vi) r s 6 (vii) r s (viii) 6 tn / s(s ) 6() (ix) 6 tn / / s(s ) 6() (x) 6 tn / / s(s ) 6() () (i) + + r r r r (ii) rrrr () If + then when 9. OTHER IMPORTNT RESULTS : () If then 60 () If 0 + then 60 () If ( + + ) ( + ) then 60 (In the right hnd side missing letter is nd the rrespnding side is 60 ) () (i) If rr r r then 90 FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

18 NRYN Mthemtis : Slutin f Tringle & Heights nd Distnes (ii) r r If r r then 90 (iii) If (r r ) (r r ) r r then 90 (iv) r + r + r r 0 then 90 (v) If R + r r then 90 (vi) If r : R : r : 5 : then 90 () () () The ve ll missing OR repeted letter tht rrespnding side is 90 If t/ : t/ : t/ x : y : z then : : y + z : z + x : x + y If x r y r z r then : : y + z : z + x : x + y (d) (r r) (r + r ) (e) r + r + r r R (f) r (r + r + r ) + + s HEIGHTS ND DISTNES () Let 'O ' e the server's eye nd OX e the hrizntl line thrugh O. () If the jet P is t higher level thn O, then ngle POX ( θ ) is lled the ngle f elevtin. () If the jet P is t lwer level thn O, then ngle POX is lled the ngle f depressin. 0. SOME USEFUL RESULTS : In tringle, () If D is medin, then + ( D + D ) FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

19 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN () If D is the ngle isetr f D D, r if D is the externl ngle isetr f then / / θ D D () If line is perpendiulr t plne, then it is perpendiulr t every line lying in tht plne. E () If DE, then DE D E θ θ D P (5) The ngle sutended y ny hrd t the entre is twie the ngle sutended y the sme n ny pint n the irumferene f the irle. α (6) ngles in the sme segment f irle re equl P Q R i.e. P Q R (7) P is sent f irle nd PT is tngent then P.P PT P T FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880 5

20 NRYN Mthemtis : Slutin f Tringle & Heights nd Distnes SOLVED EXMPLES Exmple - In, if t/ : t/ : t/ : 5 : 7 then : : () : 5 : 6 () 5 : 6 : () 6 : 5 : () 6 : : 5 Slutin : ns. () s(s ) s(s ) s(s ) : : :5:7 s : s : s : 5 : 7 s k, s 5k, s 7k s + s 8k 8k s + s k k s + s 0k 0k : : k : 0k : 8k 6 : 5 : (OR) If t/ : t/ : t/ x : y : z then : : y + z : z + x : x + y : 0 : 8 6 : 5 : Exmple - In, if 60, then () () () () Slutin : ns. () In n equilterl tringle nd 60 (OR) Exmple If 60 then + s + s If n, n +, n +, where n is ny nturl numer, represents the sides f tringle in whih the lrgest ngle is twie the smllest, then n () () () () FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

21 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN Slutin : ns. () On verifitin if n, the sides re, 5, 6 s s s s ()(5) 8 Exmple - If in tringle, 5 s + 6 s nd 6 s + s 5 then tn/ tn/ () / () / () /5 () 5 Slutin : ns. () dding the given reltins we get 9 9 s + 6 (s + s) ( + ) ( ) 9( s) 6. s s ( ) 9.sin / sin/s s / s / sin / sin / s/s/+ sin/sin/ tn/ tn/ + tn/ tn/ 5 tn/ tn/ tn/ tn/ /5 ( + ) s ( ) s tn / tn / + tn / tn / Exmple - 5 The perimeter f tringle right ngled t is 70 nd the in rdius 6 then () () () 8 () 9 Slutin : ns. () We knw tht 70 sr ( sin sin90, 90 ) 0 Nw ( + ) (70 ) ( + ) (0) ( ) FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880 7

22 NRYN Mthemtis : Slutin f Tringle & Heights nd Distnes Exmple - 6 If the re f is ( ) then its irumrdius R () sin / 6 () se / 6 () Slutin : ns. () sin / 6 ( ) + () sin / 6 + 8R s sin / s 8R 8R R se / 6 Exmple - 7 If in tringle, r r r ; D is the midpint f then s D () 7/5 () 7/5 () /5 () /5 Slutin : ns. () r r r : : 5 : : 5k, k, k + 5k is right ngled tringle with 90 sine D is the midpint f, the hyptenuse, D D D nd s D s(80 ) s s 6 7/5 5 Exmple - 8 In tringle, the sides,, re respetively,, 5. If r is the rdius f the esried irle tuhing nd the sides nd prdued then r is equl t () / () () () Slutin : ns. () s ( + + 5) s(s )(s )(s ) r s 8 8 FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

23 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN Exmple - 9 If the medin f the tringle thrugh is perpendiulr t, then tn + tn () tn () sin () s () 0 Slutin : ns. () Let E e perpendiulr t prdued, then frm similr tringles D nd E, we find E D nd E E 80 - D FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880 D tn (frm tringle D) E D D tn( ) tn s tht tn + tn 0 E Exmple - 0 If r, R, 8 then + + () 7 () 8 () 8 () 78 Slutin : ns. () We knw tht r + r + r r R r + r + r R + r 7 8 We knw tht r s 8 s r We knw tht r(r + r + r ) + + s Exmple - If r R then s + s + s () () () 5 7 () Slutin : ns. () ( + ) ( ) s + s + s s s + sin ( ) + sin/s sin / ( ) ( + ) + sin/ s s + sin/ sin/ sin/ Rsin / sin / sin / r R R r R 9

24 NRYN Mthemtis : Slutin f Tringle & Heights nd Distnes Exmple - n server n the tp f liff 00 m ve the se level serves the ngle f depressin f tw ships n ppsite sides f the liff t e 5 nd 0 respetively. The line jining the ships pints t the se f the liff. The distne etween the ships is () 00 m () 56. m () 6. m () 96. m Slutin : ns. () Let e the liff nd, D e the psitins f the ships, then 00 m D 5 nd D 0 t 5 00 m ls D t0 D m Distne etween the ships D + D 56. m Exmple - n erplne when 600 m high psses vertilly ve nther erplne t n instnt when their ngles f elevtin t the sme serving pint re 60 nd 5 respetively. The differene f the heights f the tw plnes is () 6. m () 600 m () 00 m () 5.6 m Slutin : ns. () Let e the psitin f the erplne mving 600 m high frm the hrizntl line D nd let e the psitin f nther plne t tht instnt. Let e the serving pint 5 Then D 5 nd D D D D t D ls D t 5 D D D 600 D D D m 0 FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

25 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN Exmple - spheril lln whse rdius is m, sutends n ngle α t the server s eye when the ngulr elevtin f the entre is β. The height f the entre f the lln is () rsinβ () r se α α () r () r se sinβ Slutin : ns. () Let e the entre f the lln nd O e the psitin f the server t the hrizntl line OX. Let O nd O e the tngents t the lln Exmple - 5 Then, O α β O α nd XO β nd r α O O N OX O α se O r se α / N sin O β α N O sinβ r se sinβ Sides f tringle re in.p. If < min.{, } then s is equl t () () () () Slutin : ns. () Sine sides f tringle re in.p. nd < min.{, }middle term f the.p. is either r se(i) when + + s se (ii) + s + N + ( ) + ( ) X FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

26 NRYN Mthemtis : Slutin f Tringle & Heights nd Distnes Exmple - 6 If,, e the side lengths f tringle then the vlue f the expressin is lwys () () () () Slutin : ns. () Using weighted.m. nd G.M. inequlity we get / + + Exmple If the length f medins, nd f tringle re m, m,m respetively, then () m > s () m > s 5 () m > s () m > where s is the semi-perimeter f tringle. Slutin : ns. () If G is the entrid then we hve the fllwing Similrly, G + G > m m m + m > m + m > + >, (m + m + m) > + + m + m + m > ( + + ) m + m + m > s Exmple - 8 In, s( ) + s( ) + s( ) () () ( + + ) () 0 () ( + + ) Slutin: ns. () s( ) 8R sin s( ) FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880 s

27 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN 8R sin sin( + ) s( ) R sin (sin + sin) R sin (sin s + s sin ) the given expressin R sin ( sin s + sin s ) 8R sin sin (sin s + s sin ) 8R sin sin sin( + ) 8R sin sin sin. Exmple - 9 If n ngle α is divided int tw prts nd suh tht θ, nd tn : tn m : n, then sin θ () m+ n sin α m n m () sin α m+ n m n () sin α m+ n Slutin : ns. () n () sin α m+ n + α nd θ α+θ, α θ m n tn tn α+θ tn α θ tn α+θ α θ sin s α+θ α θ s sin sinα+ sinθ sinα sinθ m n sinθ m n sinθ sinα. m + n sinα m + n Exmple - 0 In tringle where,, re ute, the distnes f the rthentre frm the sides re in the prprtin () s : s : s () sin : sin : sin () se : se : se () tn : tn : tn Slutin : ns. () HD D tn E s tn Rsinss sin Exmple Rs ss R s s s Rs ss Similrly, HE nd s HF Rs ss s HD : HE : HF s : s : s. If,, re the ngles f tringle, then sin + sin s () () () () 5 FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

28 NRYN Mthemtis : Slutin f Tringle & Heights nd Distnes Slutin: ns. () Let sin + sin s k r, + sin s + s( + ) k [ + ] r, + + sin s + sin k + + r, sin s sin + (k ) 0 + Sine this is qudrti equtin in sin whih is rel pplying sine rule in tringle O, disriminnt 0 s 8 (k ) 0 r s (k ) Exmple - r, (k ) s r k sin + sin s Prve tht the distne etween the irum-entre nd the rthentre f tringle is R 8s s s. () R + 8s ss () R 8s ss () R + 8sin sinsin () R 8sin sinsin Slutin : ns. () Let O nd P e the irumentre nd the rthentre respetively. If OF is perpendiulr t, we hve Exmple - Slutin : ns. () OF 90º OF 90º ls OP OF PL (90º ) + 80º + (+ + ) ls O R nd P R s Nw in S OP, OP O + P O.P. s OP R + R s R s s( ) R + R s { s s( )} R R s { s ( + ) + s( )} R 8R s s s Hene, OP R 8s s s. The tw djent sides f yli qudritterl re nd 5 nd the ngle etween them is 60 if the third side is. The remining furth side is () () () () 5 Let D e yli qudrilterl, 5, D, 60 D 0 FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

29 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN Exmple - In, nd D we hve + s60 r nd D + D D. D s0 r d d d + d 0 0 d + d 0 0 (d + 5) (d ) 0 d r 5 The re f the irle nd the re f the regulr plygn f n-sides nd f perimeter equl t tht f the re in the rti f () tn : n n () sin : n n Slutin : ns. () () s : n n () t : n n Let r e the rti f the nd e its re, then Sine the perimeter f the is sme is the perimeter f regulr plygn f n sides. r r n, where is the length f ne side if the regulr plygn Let e the re f the plygn. r n Then r t t n n r : r : t n n tn : n n Exmple - 5 In, / nd / nd D-divides internlly in the rti : then sin D sin D () () 6 Slutin : ns. () In D, pplying sine lw we get () () D x D x sinα sinα sin In D, pplying sine lw, we get D x D x sinβ sin sinβ...(i)...(ii) x x sinα Frm (i) nd (ii), sinα sinβ sinβ 6 5 FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

30 NRYN Mthemtis : Slutin f Tringle & Heights nd Distnes EXERISES LEVEL I. If in, r r r then : : () : : 5 () 5 : : () 5 : : () : 5 :. In n equilterl tringle r : R : r () : : () : : () : : () : :. If in tringle, 60, then () ( ) () ( ) () ( ) + () If the ngles f tringle re in.p. then () 75 () 90 () 60 () 5 5. In tringle, if, nd sin /, then () 60 () 90 () 5 () 0 6. If : : 7 : 8 : 9 then s : s : s () : : () : 7 : () : : 6 () 6 : : 7. In, f 7, 8, 9 then the length f the line jining t the midpint f is () 6 () 7 () 5 () 6 8. In, if r 6, r 8, r then the re f the tringle is () 6 () 6 () 6 () The sines f tw ngles f tringle re equl t 5/ nd 99/0. The sine f the third ngle is () 55 5 () 5 () 55 () FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

31 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN 0. Given,, 0, then inrdius f is () () + (). In right ngled tringle, sin + sin + sin () + () 0 () () (). If the ngles, nd f tringle re in P nd the sides, nd ppsite t these ngles re in G.P. ; then, nd re in () G.P. ().P. () H.P. () ll the ve. If sin sin( ), then the sides f re in ().P. () G.P. () H.P. () nne f these. If the sum f the squres f the sides f tringle is equl t twie the squre f its irum dimeter then sin + sin + sin () () () () 5. The se f tringle is 80 m nd ne f the se ngle is 60. If the sum f the lengths f the ther tw sides is 90 m then the shrtest side is () 5 m () 9 m () m () 7 m 6. In, if r 6, r 8 nd r, then the perimeter f the tringle is () 6 () 8 () 7 () 5 7. If s s, then is () isseles () right ngled () equilterl () right ngled isseles 8. If 75, 5, then + () () + + () () 9. If stnds fr the re f tringle, then sin + sin () () () () ( + + ) FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880 7

32 NRYN Mthemtis : Slutin f Tringle & Heights nd Distnes 0. If + t then the is () isseles () equilterl () right ngled () nne f these. In, the tngent f hlf the differene f tw ngles is ne-third the tngent f hlf the sum f the ngles. Then the rti f the sides ppsite t the ngles is () : () : () : () : 5. In, sin + sin + sin sin sin sin then the vlue f () 0 () (++) () (++) (++) () (++) ( ). In tringle O is pint inside the tringle suh tht O O O 5, then vlue f t + t + t is () () () + () +. On the level grund the ngle f elevtin f the tp f twer is 0. On mving 0 mt. nerer the twer, the ngle f elevtin is fund t e 60. The height f the twer is () 0 mt () 0 mt () 0 mt () nne f these 5. flg stff f 5 mt high stnds n uilding f 5 mt high. t n server t height f 0 mt, the flg stff nd the uilding sutend equl ngles. The distne f the server frm the tp f the flg stff is () 5 () 5 () 5 () nne f these 8 FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

33 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN LEVEL - II. If the ngles,, f re in.p., then () + () + () + () nne f these. In tringle, 0 + then () 0 () 5 () 60 () 90. In tringle, if the medin D mkes n ngle θ with nd D then sinθ. () sin () sin () sin () sinsin r t t () r () r () r () 5. 7 In, 5, nd tn. Then mesure f is 9 () () 7 () 6 () 5 6. R () s ( ) () sin ( ) () s s () sin sin 7. If +, then t + t t () () () 0 () 8. In, nd the mesure f is () () 6 9. Tw sides f re given y the rts f the equtin x x+ 0, the ngle etween the sides is / the perimeter f the tringle is () 6+ () + 6 () + 0 () + 9 FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880 () () 9

34 NRYN Mthemtis : Slutin f Tringle & Heights nd Distnes 0. irle is insried in n equilterl tringle f side. The re f ny squre insried in the irle is () / () /6 () /9 () /. If the se ngles f tringle re nd then the ltitude f the tringle is equl t () se () rd f se () f se () th f se. If the sides,, f re in.p., then s t, st, st ().P. () G.P. () H.P. () nne f these. In ny tringle, sin + sin + is lwys greter thn sin () 9 () () 7 () 0. sin( ) If in tringle,, then the tringle is + sin( + ) () right ngled r isseles () right ngled nd isseles () equilterl () nne f these 5. If in, s + s + s, then,, re in ().P. () H.P. () G.P. () nne f these 6. In tringle, Then the tringle is () equilterl () right-ngled nd isseles () right-ngled with 90, 60, 0 () nne f these 7. In, : : : 5 :, then + + is () () () () re in 8. In, /, m nd 9 r( ) m then is () 6 m () 9 m () 8 m () 8 m 0 FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

35 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN 9. If sin( + + ), tn( ) nd se( + ) then () 90, 60, 0 () 0, 60, 0 () 60, 0, 0 () The sides f tringle insried in given irle sutend ngle α, β nd γ t the entre. The minimum vlue f the rithmeti men f s ( α + /), s (β + /) nd s ( γ + /) is equl t () 0 () / () () /. If the sides f right ngled tringle re in.p. then the tngents f the ute ngled tringled re () (), +, () (), 5 +, 5. If,, re given in, nd, re the pssile vlues f the third side, then + s () sin () s () sin () s. The sides f tringle re x+y, x+y nd 5x+5y units, where x, y > 0. The tringle is () right ngled () equilterl () tuse ngled () isseles. n erplne flying t height f 00 m ve the grund psses vertilly ve nther plne t n instnt when the ngles f elevtin f the tw plnes frm the sme pint n the grund re 60 nd 5 respetively. The height f the lwer plne frm the grund is () 00 m () 00 / m () 50 m () 50( + ) m 5. Tw sides f tringle re m nd m nd the ngle ppsite t the shrter side f the tw is the lrgest pssile length f the third side is () ( ± )m () (6 ± )m () ( 6 ± )m () 6 ± FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

36 NRYN Mthemtis : Slutin f Tringle & Heights nd Distnes LEVEL - III. In, s /5 nd s 5/ then s () 7 () () 65 () 6 5. The length f the sides f tringle re x, y nd x + y + xy the mesure f the gretest ngle is () () 5 6 () () 5. In, the sides re in the rti : 5 : 6 the rti f the irumrdius nd inrdius is () 8 : 7 () : () 7 : () 6 : 7. If s + s + s then the sides f re in ().P () G.P () H.P ().G.P 5. In, the sides, nd re suh tht they re the rts f x x + 8x 0 0 then s s s + + () () 9 6 () 6. The sides f tringle re in the rti : 6: +, then its ngles re () 5,5,90 () () 9 60,0,90 () 5,60,75 () nne f these 7. If in, + then tn/ tn/ () tn/ () () () / 8. + In ny s( )s + () + + () + + () + + () nne f these FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

37 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN 9. If 90 in the tringle, then tn + + tn + () () () 6 0. The re f is ( ), then tn is equl t () () () () () 8 5. If in s s s then the vlue f the ngle is () () (). If the sides f tringle re in.p. nd is the smllest side then s in terms f, is () 6 () + () () + () +. In tringle ( + + ) ( + ) the mesure f ngle is () () 6 () (). The equtin x + x + 0, where,, re the sides f nd the equtin x + x+ 0 hve mmn rt, the mesure f is () 90 () 5 () 60 () 0 5. If the rdius f the irumirle f n isseles PQR is equl t PQ( PR) then P () /6 () / () / () / FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

38 NRYN Mthemtis : Slutin f Tringle & Heights nd Distnes 6. In, nd the perimeter is six times the.m f the sines f the ngles the mesure f is () () 6 () () 7. If,,, re res f exirles nd inirles f tringle, then + + () () () () nne 8. The re f the tringle whse sides re +, +, + where,, > 0 is () + + () + + () + + () ( + + ) 9. If αβγ,, re the lengths f the ltitudes f, then α +β + γ () () t + t + t () t + t + t () t + t + t 0. In tringle, 5, 7 nd sin, hw mny suh tringles re pssile () () 0 () () infinite. If in tringle, ( + + ) ( + ) λ then exhustive rnge f λ is () (, ) () (0, ) () (0, ) () (, ). In tringle the vlue f the expressin r r + r r + r r is lwys equl t () ( + + ) () ( ) () ( + + ) () ( + + ) FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

39 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN. If dentes the re f ny tringle nd S its semiperimeter, then () < s () > s s () < () nne f these. mn in t rwing wy frm liff 50 metres high serves tht it tkes minutes t hnge the nlge f elevtin f the tp f the liff frm 60 t 5. The speed f the t is () (/ ) (9 ) km/h () (/ ) (9 + ) km/h () (/)(9 )km/h () nne f these 5. If p, p, p re the lengths f the ltitudes f tringle frm the verties,,, then s s s + + p p p () s / ( + + ) () R () t + t + t () R FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880 5

40 NRYN Mthemtis : Slutin f Tringle & Heights nd Distnes QUESTIONS SKED IN IEEE & OTHER ENGINEERING EXMS. twer stnds t the entre f irulr prk. nd re tw pints n the undry f the prk suh tht ( ) sutends n ngle 60 t the ft f the twer, nd the ngle f elevtin f the tp f the twer frm r is 0. THe height f the twer is () / () () / () [IEEE - 007]. In tringle, let. If r is the inrdius nd R is the irumrdius f the tringle, then (r + R) equls () + () + () + + () + [IEEE - 005]. If in, the ltitudes frm the verties,, n ppsite sides re in H.P., then sin, sin, sin re in () G.P. ().P. ().G.P. () H.P. [IEEE - 005]. The sides f tringle re sinα, sα nd + sinαsα fr sme 0 <α< /. Then the gretest ngle f the tringle is () 90 () 60 () 0 () 50 [IEEE - 00] 5. persn stnding n the nk f river serves tht the ngle f elevtin f the tp f tree n the ppsite nk f the river is 60 nd when he retires 0 metres wy frm the tree, the ngle f elevtin emes 0. The redth f the river is () 0m () 0m () 0m () 60 m [IEEE - 00] 6. th The upper prtin f vertil ple sutends n ngle tn t pint in the hrizntl plne 5 thrugh its ft nd t distne 0m frm the ft. pssile height f the vertil ple is () 0m () 0m () 60m () 80m [IEEE - 00] 7. In tringle, medins D re E re drwn. If D, D / 6 nd E /, then the re f the is () 6 / () / () / () 6/ [IEEE - 00] 8. If in tringle, s + s then the sides,, nd () re in.p. () re in G.P. () re in H.P. () stisfy + [IEEE - 00] 6 FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

41 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN 9. The sum f the rdii f insried nd irumsried irles fr n n sided regulr plygn f side '' is () t n () t n () t n () t n [IEEE - 00] 0. In tringle,,, 60, then is the rt f the equtin () 7 0 () () () [IEEE - 00]. 5 In,tn,tn, then 6 5 (),, re in.p. (),, re in.p. (),, re in.p. () nne f these [IEEE - 00]. Let T n dente the numer f tringles whih n e frmed using the verties f regulr plygn f n sides. If T n+ T n then n () 5 () 7 () 6 () [IEEE - 00]. + In tringle, sin is equl t () + () + () () [IEEE - 00]. The sides f tringle re, 5 nd 6 m. The re f the tringle is equl t () 5 m () 5 7m () 7m 5 () nne f these [UPSET - 00] 5. In tringle if, 0 then the re f the irumirle f tringle in squre units is () () () () 6 [ET (Krntk) - 00] 6. If R is the rdius f the irumirle f the, nd is its re then () + + R () + + R () R () R [ET (Krntk) - 000] 7. Let the ngles,, f e in.p. nd let : :. Then ngle is () 75 () 5 () 60 () nne f these [EET (Hryn) - 000] 8. In, if s s then tringle is () right ngled () isseles () equilterl () nne f these [EET (Hryn) - 000] 7 FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

42 NRYN Mthemtis : Slutin f Tringle & Heights nd Distnes 9. If in tringle, D, E nd F re the ltitudes nd R is the irum rdius, then the rdius f the irle DEF is () R () R () R () nne f these [EET (Hryn) - 000] 0. If D is the mid pint f the side f tringle nd D is perpendiulr t, then () () + 5 () () [EET (Hryn) - 999]. s + s + s () r () R r R () r + () nne f these [EET (Hryn) - 999] R. The perimeter f is 6 times the rithmeti men f the sines f its ngles. If the side is, then the ngle is () 6 () () () [EET (Hryn) - 998]. If r, r, r in tringle e in H.P., then the sides re in () H.P. ().P. () G.P. () nne f these [EET (Delhi) - 000]. If ( ) seθ, then sin () sθ () tθ () tnθ () sinθ [EET (Delhi) - 000] s s s 5. In, nd. re f the tringle is () () () () [PET (MP) - 000] 8 FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880

43 Mthemtis : Slutin f Tringle & Heights nd Distnes NRYN NSWERS EXERISES LEVEL I. (). (). (). () 5. () 6. () 7. () 8. () 9. () 0. (). (). (). (). () 5. () 6. () 7. () 8. () 9. () 0. (). (). (). (). () 5. () LEVEL II. (). (). (). () 5. () 6. () 7. () 8. () 9. () 0. (). (). (). (). () 5. () 6. () 7. () 8. () 9. () 0. (). (). (). (). () 5. () LEVEL - III. (). (). (). () 5. () 6. () 7. () 8. () 9. () 0. (). (). (). (). () 5. () 6. () 7. () 8. () 9. () 0. (). (). (). (). () 5. () QUESTIONS SKED IN IEEE & OTHER ENGINEERING EXMS. (). (). (). () 5. () 6. () 7. () 8. () 9. () 0. (). (). (). (). () 5. () 6. () 7. () 8. () 9. () 0. (). (). (). (). () 5. () 9 FNS Huse, 6, Klu Sri Mrket, Srvpriy Vihr, New Delhi-006! Ph.: (0) 00/ Fx : (0) 880