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M`htchts`htchts [rnl`h`dctry;.6 Wnlet-fhlick [rnfhlics 7

;.7 [rnl`h`dctrnm Buhmtn`hs 6>

;.= [rnl`h`dctrnm Nkchtntncs =2

;.; Fppinmftn`hs `b [rnl`h`dctry t` [rnfhlics 3=

;.3 Fppinmftn`hs `b [rnl`h`dctry t` \fvcs 23

Icfrhnhl

Nh tens \`rog``o y`u wnii icfrh fg`ut tec gfsnm guniknhl gi`mos `b trnl`h`dctry. ^`u wnii

icfrh fg`ut tec snhc, m`snhc, tfhlcht, m`scmfht, scmfht, m`tfhlcht buhmtn`hs fhk tecnr

dfhy ndp`rtfht rciftn`hsenps. ^`u wnii icfrh fg`ut tecnr lrfpes fhk tecnr pcrn`knm hfturc.

^`u wnii icfrh e`w t` fppiy Qytefl`rfs' tec`rcd fhk tec ]nhc fhk M`snhc ruics t` bnhk

ichltes fhk fhlics `b trnfhlics.

`utm`dcs

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Wnlet-fhlick

[rnfhlics

;.6Nhtr`kumtn`h

Wnlet-fhlick trnfhlics (teft ns trnfhlics wecrc `hc `b tec fhlics ns >1) frc tec cfsncst t`pnm b`rnhtr`kumnhl trnl`h`dctry. ]nhmc tec sud `b tec tercc fhlics nh f trnfhlic ns 651 nt b`ii`ws teft nhf rnlet-fhlick trnfhlic tecrc frc h` `gtusc fhlics (n.c. fhlics lrcftcr tefh >1). Nh tens ]cmtn`h wcstuky dfhy `b tec pr`pcrtncs fss`mnftck wnte rnlet-fhlick trnfhlics.

Qrcrcqunsntcs

Gcb`rc stfrtnhl tens ]cmtn`h y`u se`uik. . .

efvc f gfsnm oh`wicklc `b tec lc`dctry `btrnfhlics

Icfrhnhl @utm`dcs

@h m`dpictn`h y`u se`uik gc fgic t`. . .

kcsthc trnl`h`dctrnm buhmtn`hs g`te nhrnlet-fhlick trnfhlics fhk d`rc lchcrfiiy

cxprcss fhlics nh kclrccs `r rfknfhs

mfimuiftc fii tec fhlics fhk snkcs nh fhyrnlet-fhlick trnfhlic lnvch mcrtfnh nhb`rdftn`h

7 ECID (7113):\`rog``o ;: [rnl`h`dctry

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6. Wnlet-fhlick trnfhlicsI``o ft Bnlurc 6 wenme m`uik, b`r cxfdpic, gc f pr`stic `b f enii wnte f m`hstfht lrfkncht.

F M6 M7

G6

G7

F

Bnlurc 6

[ec tw` rnlet-fhlick trnfhlicsFG6M6 fhkFG7M7 frcsndnifr(gcmfusc tec tercc fhlics `b trnfhlicFG6M6frc cqufi t` tec cqunvficht = fhlics `b trnfhlicFG7M7). Br`d tec gfsnm pr`pcrtncs `b sndnifrtrnfhlics m`rrcsp`hknhl snkcs efvc tec sfdc rftn`. [eus, b`r cxfdpic,

G6M6FG6

4G7M7FG7

fhk FM6

FG64

FM7FG7

(6)

[ec vfiucs `b tec tw` rftn`s (6) wnii micfriy kcpchk `h tec fhlicF `b nhminhftn`h. [ecsc rftn`s frcmfiick tec snhc fhk m`snhc `b tec fhlicF, tecsc gcnhl fggrcvnftck t` snh Ffhk m`s F.

Ocy Q`nht 6

F M

G

F

G

snh F4GM

FG m`s F4

FM

FG

Bnlurc 7

FM ns tec snkc fkjfmcht t` fhlicF.

GM ns tec snkc `pp`sntc t` fhlicF.

FG ns tec eyp`tchusc `b tec trnfhlic (tec i`hlcst snkc).

[fsofsoWcbcrrnhl flfnh t` Bnlurc 7 nh Ocy Q`nht 6, wrntc k`wh tec rftn`s wenme lnvc snh Gfhk m`s G.

^`ur s`iutn`h

Fhswcr

snh G 4

FM

FG m`s G 4

GM

FG .H`tc teftsnh G 4 m`s F4 m`s(>1 G) fhk m`s G 4 snh F4 snh(>1 G)

ECID (7113):]cmtn`h ;.6: Wnlet-fhlick [rnfhlics

=

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F tenrk rcsuit `b ndp`rtfhmc br`d Bnlurc 6 ns

G6M6FM6

4G7M7FM7

(7)

[ecsc rftn`s ns rcbcrrck t` fs tec tfhlcht `b tec fhlic ftF, wrnttch tfh F.

Ocy Q`nht 7

F M

G

F

G

snh F4GM

FG m`s F4

FM

FG

Bnlurc =

tfh F4GM

FM 4

ichlte `b `pp`sntc snkc

ichlte `b fkjfmcht snkc

B`r fhy rnlet-fhlick trnfhlic tec vfiucs `b snhc, m`snhc fhk tfhlcht frc lnvch nh Ocy Q`nht =.

Ocy Q`nht =

F M

G

F

G

snh F4GM

FG m`s F4

FM

FG

Bnlurc ;

\c mfh wrntc, tecrcb`rc, b`r fhy rnlet-fhlick trnfhlic m`htfnhnhl fh fhlic (h`t tec rnlet-fhlic)

snh 4 ichlte `b snkc `pp`sntc fhlic

ichlte `b eyp`tchusc 4@pp

Eyp

m`s 4 ichlte `b snkc fkjfmcht t` fhlic

ichlte `b eyp`tchusc 4

Fkj

Eyp

tfh 4 ichlte `b snkc `pp`sntc fhlic

ichlte `b snkc fkjfmcht t` fhlic4

@pp

Fkj

[ecsc frc s`dctndcs dcd`rnsck fs]@E,MFE fhk[@F rcspcmtnvciy.

[ecsc tercc rftn`s frc mfiick trnl`h`dctrnm rftn`s.

; ECID (7113):\`rog``o ;: [rnl`h`dctry

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[fsofso\rntctfh nh tcrds `bsnh fhk m`s .

^`ur s`iutn`h

Fhswcr

tfh 4@pp

Fkj 4

@pp

Fkj.

Eyp

Eyp4

@pp

Eyp.Eyp

Fkj 4

@pp

Eyp

FkjEyp

n.c. tfh 4 snh

m`s

Ocy Q`nht ;

Qytefl`rfs [ec`rcd

f

gmf7 +g7 4m7

Bnlurc 3

Cxfdpic 6Ssc tec ns`smcics trnfhlic nh Bnlurc 2 t` `gtfnh tec snhc, m`snhc fhk tfhlcht `b;3.

F M

;3

G

;3

x

x

Bnlurc 2

]`iutn`h

Gy Qytefl`rfs tec`rcd (FG)7 4x7 +x7 4 7x7 s` FG4x

7

Echmc snh ;34

GM

FG 4

x

x7 4 6

7 m`s;3 4FM

FG 4

6

7 tfh;34GM

FM 4

x

x 4 6

ECID (7113):]cmtn`h ;.6: Wnlet-fhlick [rnfhlics

3

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Chlnhccrnhl Cxfdpic 6

H`nsc rckumtn`h gy s`uhk gfrrncrs

Nhtr`kumtn`h

Fukngic s`uhk efs dume i`hlcr wfvcichltes tefh inlet. M`hscquchtiy, s`uhk trfvciinhl nh tec ftd`-specrc ns fgic t` gchk fr`uhk `gstfmics cvch wech tecsc `gstfmics mfusc sefrp sefk`ws b`r inlet.[ens ns tec rcsuit `b tec wfvc pech`dch`h oh`wh fs knfirfmtn`h. Nt mfh gc `gscrvck fis` wnte wftcrwfvcs ft tec chks `b grcfowftcrs. [ec cxtcht t` wenme wfvcs gchk fr`uhk `gstfmics kcpchks up`htec wfvcichlte fhk tec s`urmc-rcmcnvcr lc`dctry. ]` tec cmfmy `b purp`sc gunit h`nsc gfrrncrs, sumefs t` gc b`uhk fi`hlsnkc d`t`rwfys nh urgfh fhk sugurgfh frcfs, kcpchks `h tec brcquchmncs nh tecs`uhk fhk tec i`mftn`hs `b tec s`urmc fhk rcmcnvcr (hcfrcst h`nsc-fficmtck pcrs`h `r kwciinhl) rciftnvc

t` tec gfrrncr. ]pcmnstmfiiy, tec gfrrncr pcrb`rdfhmc kcpchks `h tec knficrchmc nh tec ichltes `b teceyp`tectnmfi rfy pftes pfssnhl br`d s`urmc t` rcmcnvcr cntecr knrcmtiy `r vnf tec t`p `b tec gfrrncr (sccBnlurc

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Dftecdftnmfi fhfiysns

H`tc tec ifgcisR,S, \ `h p`nhts teft frc uscbui b`r tec fhfiysns. H`tc teft tec ichlte `bWR 4er es fhk teft tec e`rnz`htfi scpfrftn`h gctwcch] fhkW nsr+s. Nh tec rnlet-fhlick trnfhlic]WR, Qytefl`rfs tec`rcd lnvcs

(]W)7 4 (r+s)7 + (er es)7

]`

]W 4

(r+s)7 + (er es)7 (=)H`tc teft tec ichlte `b[S4E esfhk tec ichlte `b[\ 4E er.Nh tec rnlet-fhlick trnfhlic][S,

(][)7 4s7 + (E es)7

Nh tec rnlet-fhlick trnfhlic [\W,

([W)7

4r7

+ (E er)7

]`

][+[W4

s7 + (E es)7 +

r7 + (E er)7 (;)]` usnhl (=) fhk (;)

][+[W ]W 4

s7 + (E es)7 +

r7 + (E er)7

(r+s)7 + (er es)7.B`res4 6, s4 =, E4 =, r4 =1 fhker4 6,

][+[W

]W 4 =7 + (= 6)7 + =17 + (=

6)7

(=1 + =)7 + (6

6)7

4

6= +

>1; ==

4 1.2

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Chlnhccrnhl Cxfdpic 7

E`rnz`h knstfhmc

Qr`gicd nh w`rks

I``onhl br`d f ecnlet `b 7 d fg`vc scf icvci, e`w bfr fwfy ns tec e`rnz`h0 ]tftc fhy fssudptn`hsdfkc.

Dftecdftnmfi stftcdcht `b tec pr`gicd

Fssudc teft tec Cfrte ns f specrc. Bnhk tec ichlteK`b tec tfhlcht t` tec Cfrtes specrc br`d tec`gscrvftn`h p`nht@.

W

e

K

W@

Bnlurc 5: [ec Cfrtes specrc fhk tec tfhlcht br`d tec `gscrvftn`h p`nht@

Dftecdftnmfi fhfiysns

Ssnhl Qytefl`rfs tec`rcd nh tec trnfhlic se`wh nh Bnlurc 5,

(W+e)7 4K7 +W7

Echmc

W7 + 7We+e7 4K7 +W7 e(7W+e) 4K7 K4

e(7W+e)

NbW4 2.=.

1 7 ; 2 5 61

3111

61111

63111

Ecnlet d

E`rnz`hK

Bnlurc >

5 ECID (7113):\`rog``o ;: [rnl`h`dctry

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Ft fh `gscrvftn`h ecnlet `b 7 d, tec b`rduif prcknmts teft tec e`rnz`h ns just `vcr 3 od fwfy. Nhbfmt tec vfrnftn`h `b `ptnmfi rcbrfmtnvc nhkcx wnte ecnlet nh tec ftd`specrc dcfhs teft tec e`rnz`h nsfppr`xndftciy >% lrcftcr tefh tens.

[fsofsoSsnhl tec trnfhlicFGM nh Bnlurc 61 wenme mfh gc rclfrkck fs `hc efib `b teccquniftcrfi trnfhlicFGK, mfimuiftc snh, m`s, tfh b`r tec fhlics =1 fhk 21.

F M

G

x

K

=1

21

x

7

Bnlurc 61

^`ur s`iutn`h

Fhswcr

Gy Qytefl`rfs tec`rcd: (GM)7 4 (FG)7 (FM)7 4x7 x7

; 4

=x7

; s` GM4x

=

7

Echmc snh214GM

FG4

x=

7

x 4

=

7 snh=14

FM

FG 4

x

7

x4

6

7 m`s21 4

FM

FG4

6

7

m`s=14GM

FG 4

x=

7

x 4

=

7 tfh21 4

=

7

6

7

4

= tfh =146

7=

7

4 6

=

Rfiucs `bsnh , m`s fhk tfh mfh `b m`ursc gc `gtfnhck gy mfimuift`r. \ech chtcrnhl tec fhlicnh kclrccs ( c.l. =1) tec mfimuift`r dust gc nh kclrcc d`kc. ([ypnmfiiy tens ns chsurck gy prcssnhltec KWL gutt`h uhtni KCL ns se`wh `h tec knspify). [ec ocystr`ocs b`r snh =1 frc usufiiy sndpiysnh =1 `r, `h s`dc mfimuift`rs, =1 snh pcrefps b`ii`wck gy 4.

[fsofso(f) Ssc y`ur mfimuift`r t` mecmo tec vfiucs `bsnh ;3, m`s=1fhk tfh 21`gtfnhck

nh tec prcvn`us [fso.

(g) Fis` `gtfnh snh=.7, m`s52.5, tfh7563. ( kch`tcs f dnhutc 4 6

21

)

ECID (7113):]cmtn`h ;.6: Wnlet-fhlick [rnfhlics

>

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^`ur s`iutn`h

(f)

(g)

Fhswcr(f) 1.

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[fsofsoMecmo tec vfiucs `b tec fhlics ftF fhkG nh Bnlurc 66 fg`vc usnhl tec m`s6

buhmtn`hs `h y`ur mfimuift`r. Lnvc y`ur fhswcrs nh kclrccs t` 7 k.p.

^`ur s`iutn`h

Fhswcr

F4 m`s6;

3 4 =2.5

8/13/2019 4 1 Right Angled Triangles

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[fsofsoSsc y`ur mfimuift`r t` `gtfnh t` = k.p. m`scm =5.3, scm 77.2, m`t55.=7 (Ssctec snh, m`s `r tfh gutt`hs uhicss y`ur mfimuift`r efs spcmnstm gutt`hs.)

^`ur s`iutn`h

Fhswcr

m`scm =5.34 6

snh=5.34 6.212 scm 77.2 4

6

m`s 77.24 6.15=

m`t55.=74 6

tfh55.=74 1.17>

7. ]`ivnhl rnlet-fhlick trnfhlics]`ivnhl rnlet-fhlick trnfhlics dcfhs `gtfnhnhl tec vfiucs `b fii tec fhlics fhk fii tec snkcs `b f lnvchrnlet-fhlick trnfhlic usnhl tec trnl`h`dctrnm buhmtn`hs (fhk, nb hcmcssfry, tec nhvcrsc trnl`h`dctrnmbuhmtn`hs) fhk pcrefps Qytefl`rfs tec`rcd.[ecrc frc tercc mfscs t` gc m`hsnkcrck:

Mfsc 6 Lnvch tec eyp`tchusc fhk fh fhlic

\c usc snh `r m`s fs fppr`prnftc:

x

ye

F

G

M

(f)

Bnlurc 67

Fssudnhlefhk nh Bnlurc 67 frc lnvch tech

m`s 4x

ewenme lnvcsx4em`s

br`d wenmexmfh gc mfimuiftck.

Fis`

snh 4

y

e s`y4esnh wenme chfgics us t` mfimuiftcy.Micfriy tec tenrk fhlic `b tens trnfhlic (ftG) ns>1 .

67 ECID (7113):\`rog``o ;: [rnl`h`dctry

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Mfsc 7 Lnvch f snkc `tecr tefh tec eyp`tchusc fhk fh fhlic.

\c usc tfh:

(f) Nbxfhk frc oh`wh tech, nh Bnlurc 67, tfh 4y

x s` y 4x tfh

wenme chfgics us t` mfimuiftcy.

(g) Nbyfhkfrc oh`wh tech tfh 4y

x lnvcsx4

y

tfh br`d wenmexmfh gc mfimuiftck.

[ech tec eyp`tchusc mfh gc mfimuiftck usnhl Qytefl`rfs tec`rcd: e4

x7 +y7

Mfsc = Lnvch tw` `b tec snkcs

\c usc tfh6 `rsnh6 `rm`s6:

(f)

x

y tfh 4y

x s` 4 tfh6

yx

Bnlurc 6=

(g)

ye

snh 4y

e s` 4 snh6

ye

Bnlurc 6;

snh 4y

e s` 4 snh6

ye

(m)

x

em`s 4

x

e s` 4 m`s6

xe

Bnlurc 63

H`tc: snhmc tw` snkcs frc lnvch wc mfh usc Qytefl`rfs tec`rcd t` `gtfnh tec ichlte `b tec tenrk

snkc ft tec `utsct.

ECID (7113):]cmtn`h ;.6: Wnlet-fhlick [rnfhlics

6=

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Chlnhccrnhl Cxfdpic =

Rnhtflc mfr grfoc pckfi dcmefhnsd

Nhtr`kumtn`h

Bnlurc 62 se`ws tec strumturc fhk s`dc kndchsn`hs `b f vnhtflc mfr grfoc pckfi frrfhlcdcht fsbfr fs tec grfoc mfgic. [ecd`dcht `b f b`rmc fg`ut f p`nht ns tec pr`kumt `b tec b`rmc fhk tecpcrpchknmuifr knstfhmc br`d tec p`nht t` tec inhc `b fmtn`h `b tec b`rmc. [ec pckfi ns pnv`tck fg`uttec p`nhtF. [ec d`dchts fg`utFdust gc cqufi fs tec pckfi ns stftn`hfry.

Qr`gicd nh w`rks

Nb tec krnvcr suppincs f b`rmc `b >11H, t` fmt ft p`nhtG, mfimuiftc tec b`rmc (B) nh tec mfgic.

Dftecdftnmfi stftcdcht `b pr`gicd

[ec pcrpchknmuifr knstfhmc br`d tec inhc `b fmtn`h `b tec b`rmc pr`vnkck gy tec krnvcr t` tec pnv`tp`nhtFns kch`tck gyx6 fhk tec pcrpchknmuifr knstfhmc br`d tec inhc `b fmtn`h `b b`rmc nh tec mfgict` tec pnv`t p`nhtFns kch`tck gyx7. Ssc trnl`h`dctry t` rciftcx6fhkx7t` tec lnvch kndchsn`hs.Mfimuiftc mi`mownsc fhk fhtnmi`mownsc d`dchts fg`ut tec pnv`t fhk sct tecd cqufi.

63

x7

x6

;1B G

F

mfgic

11 H

;1

Bnlurc 62: ]trumturc fhk kndchsn`hs `b vnhtflc mfr grfoc pckfi frrfhlcdcht

Dftecdftnmfi Fhfiysns

[ec knstfhmcx6 ns b`uhk gy m`hsnkcrnhl tec rnlet-fhlick trnfhlic se`wh nh Bnlurc 6< fhk usnhl teckcsthntn`h `b m`snhc.

x6

761 dd

;1m`s(;1) 4

x61.761

x64 626 dd.echmc

Bnlurc 6