4 1 right angled triangles
TRANSCRIPT
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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
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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