Grade 10 Maths 13

fkdf,a fnod yeu i`oyd h' fkdf,a fnod yeu i`oyd h'

fuu mdvu yeoEfuka Tng

jD;a;hl fldaK inkaO m%fh y`kd .ekSug yd tAjd Nd; lsug

yelsshdj ,efnkq we;'

31'1 jD;a; pdmhlska jD;a;fha flakah u; iy jD;a;fha mh u;

wdmd;kh flfrk fldaK

uyd pdmh

iq`M pdmhA

O

B

by; jD;a;h u; msys A iy B ,CIH foflka jD;a;h fldgia follg fjkaf' fuu fldgiaj,g pdm hehs lshkq ,ef' A yd B hd lrk fLdj jD;a;fha flakah yryd .uka lrk g" tk jD;a;fha IalNhla jk g fuu pdm fol .ska iudk f' tfia .uka fkdlrk g pdm fol .ska wiudk f' fug .ska wvq pdmhg iq pdmh hehs o .ska je pdmhg uyd pdmh hehs o lshkq ,ef'

O

BA

by; rEmfha ;o mdka oelafjk iq pdmfha fofl


(i) jD;a;fha b;s fldgi u; wdmd;kh flfrk fldaKh;a (ii) jD;a;fha flakah u; wdmd;kh flfrk fldaKh;a ,shd olajkak' â& î& P îi& ^b& î& PQR

îi& mrdj;k fldaKh

311 wNHdih

1' we;s rEm igyfka oelafjk jD;a; y;r Tf wNHdi fmdf;a msgm;a lr .kak' O u.ska oelafjkafka tla tla jD;a;fha flakahhs'

B

a

o

A B

b

o

A B

c

o

A B

d

o

A

my; oelafjk tla tla wjia:df wid we;s fldaKh ,l=Kq lrkak'

î& a rEmfha iq pdmh u.ska jD;a;fha b;s pdm fldgi u; wdmd;kh flfrk fldaKhla

îi& b rEmfha iq pdmh u.ska flakah u; wdmd;kh flfrk fldaKh ^ iii& c rEmfha uyd pdmh u.ska jD;a;fha b;s pdm fldgi u; wdmd;kh flfrk

fldaKhla

îv& d rEmfha uyd pdmh u.ska flakah u; wdmd;kh flfrk fldaKh2' rEmigyk wkqj"

Y

B

X

A

O

î& AB iq pdmh u.ska â& jD;a;fha b;s pdm fldgi u; wdmd;kh flfrk

fldaKhla

^b& jD;a;fha flakah u; wdmd;kh flfrk fldaKh ,shkak'

îi& AB uyd pdmh u.ska â& jD;a;fha b;s pdm fldgi u; wdmd;kh flfrk fldaKhla ^b& jD;a;fha flakah u; wdmd;kh flfrk fldaKh ,shkak'

3' we;s rEmfha jD;a;fha flakah O f' PQ uyd pdmh u; X ,laIH msysgd we;'

O

Q

P

X

î& PQ iq pdmh u.ska jD;a;fha b;s pdm fldgi u; wdmd;kh flfrk fldaKh ,shkak'

îi& PQ iq pdmh u.ska jD;a;fha flakah u; wdmd;kh flfrk fldaKh ,shkak'

îii& PQ iq pdmh u; kEu ,CIHhla ,l=Kq lr th Y f,i k lrkak' PYQ fldaKh y`kajkak'

îv& PQ uyd pdmh u.ska jD;a;fha flakah u; wdmd;kh flfrk fldaKh ,shkak'

4' we;s rEmfha oelafjk jD;a;fha flakah O f'î& AC iq pdmh u.ska jD;a;fha b;s pdm fldgi u; wdmd;kh flfrk

fldaKhla k lrkak'

îi& AC iq pdmh flakah u; wdmd;kh flfrk fldaKh

OC

D

BA

,shkak'

îii& AC uyd pdmh u.ska jD;a;fha b;s pdm fldgi u; wdmd;kh flfrk fldaKhla ,shkak'

îv& AC uyd pdmh u.ska jD;a;fha flakah u; wdmd;kh flfrk fldaKh ,shkak'

5' we;s rEmfha oelafjk jD;a;fha flakah O f'

O

C

B

A

î) AB iq pdmh u.ska â& jD;a;fha b;s pdm fldgi u; wdmd;kh flfrk fldaKh

^b& jD;a;fha flakah u; wdmd;kh flfrk fldaKh ,shkak'

îi& BC iq`M pdmh u.ska â& jD;a;fha b;s pdm fldgi u; wdmd;kh flfrk fldaKh ^b& jD;a;fha flakah u; wdmd;kh flfrk fldaKh ,shkak'

31.2 pdmhlska flakah yd jD;a;h u; wdmd;kh lrk fldaK w;r inkaOh

jD;a; pdmhlska flakah u; wdmd;kh flfrk fldaKh iy tu pdmh u.ska jD;a;fha b;s pdm fldgi u; wdmd;kh flfrk fldaKh w;r inkaOh ms


l%shdldrlu

IQ lvodishl jD;a;hla we`o tys flakah O f,i k lrkak' iq pdmhla yd uyd pdmhla ,efnk m jD;a;h u; ,CIH folla ,l=Kq lrkak' tu ,CIH A yd B f,i k lrkak'

X

o

BA

uyd pdmh u; ,CIHhla ,l=K lr th X f,i k lrkak'

AB iq pdmfhka flakah u; wdmd;kh flfrk fldaKh y`kd .kak' th AOBf' my; rEm igyfka oelafjk m AOB flakasl LKavh lmd fjkalr .kak'

X

O

O

A,BC

B

BA

AOB y wvla ,nd .ekSu i`oyd fjka lr.;a AOB flakasl LKavfha OA u; OB imd; jk fia folg kjkak' kjk ,o flakasl LKAvh fldaKh u; ;nd ksCIKh lrkak'

tk" AB iq`M pdmh u.ska jD;a;fha flakah u; wdmd;kh flfrk fldaKh jk AOB tu pdmh u.ska jD;a;fha b;s pdm fldgi u; wdmd;kh lrk fuka fo.=Khla njg ;yjqre jkakg we;' by; wdldrhgu fjkia wrj,ska hq;a jD;a;j, fjkia .ska hq;a pdm fldgia ,l=Kq lr" by; ls%hdldrlu kej; lrkak'

by; l%shdldrlj, jD;a;hl pdmhlska flakah u; wdmd;kh flfrk fldaKh tu pdmh u.ska jD;a;fha b;s pdm fldgi u; wdmd;kh flfrk fldaKh fuka fo.=Khla nj ksCIKh lsug yelsjkq we;'

tu m%;sM,h cHd;sl m%fhhla f,i my; oelaf'

m%fhh

jD;a; pdmhla u.ska jD;a;fha flakah u; wdmd;kh flfrk fldaKh" tu pdmh u.ska jD;a;fha b;s fldgi u; wdmd;kh flfrk fldaKh fuka fo.=Khla f'

by; oelajq m%fhh Nd;fhka .Kkh ls lrk whqre my; oelafjk ksoiqka u.ska uid n,uq'

flakah O jk jD;a;hla u; A" B iy C f,i ,laIH msysgd we;' OAB = 20x k ACB w.h fidhuq'

O

C

B

A20x

OA = OB ^tlu jD;a;fha wrh iudk f'&OAB ;%sfldaKh iuoamdo ;%sfldaKhla f'

iuoamdo ;%sfldaKhl iudk mdoj,g iuqL fldaK iudk ksid

AB iq`M pdmh u.ska flakah u; wdmd;kh flfrk fldaKh f' tu pdmh u.ska jD;a;fha b;s pdmh u; wdmd;kh flfrk fldaKh ACB ksid' m%fhh wkqj

166 167


ksoiqk 1

rEm igyfka oelafjk jD;a;fha flakah O f' we;s f;dr;=re weiqka PQRfidhkak'

O

P

RQ

15x

50x

^jD;a;hl pdmhlska flakah u; wdmd;kh flfrk fldaKh tu pdmfhka jD;a;fha b;s pdm fldgi u; wdmd;kh flfrk fldaKh fuka fo.=Khla f'&

50

312 wNHdih

1' my; oelafjk tla tla jD;a;fhys flakah O u.ska oelaf' we;s o;a; wkqj x ys w.h fidhkak'

70x

x

O

(vi)

80x

xO

(v)

50x

x

O

(iv)

50x

x

O

(iii)

30x

x O

(ii)

100x

x

O

(i)

2' my; oelafjk tla tla jD;a;fha flakah O u.ska oelaf' we;s o;a; wkqj" fya;= olajka y ys w.h fidhkak'

(i)

o

y

P

Q

R

S

100x

îi&

o

y

A B

C

îii&

6070,

oyB

A

C

îv& 30x

20x

yO

A

C

B ^v&

oy

P R

Q110x

^vi&

oy

A

BC

30x 110x

168 169


^vii&

oy

A

BC

70x ^viii&

o

y

Q

P R50x

^ix&

o

y

C

B

A

130x

31'3 jD;a;hl pdmhlska flakah u; wdmd;kh lrk fldaKh tu pdmfha b;s fldgi u; wdmd;kh lrk fldaKh fuka fo.=Khla fZZ hk m%fhfha u;a idOkh

O O

C

B BC

A AX

X

o;a;h: O flakah jQ jD;a;h u; A"B iy C ,laIH msysgd we;' idOkh l< hq;a;: nj ksudKh: CO fLdj X olajd la lsu idOkh: ^tlu jD;a;fha wr&

1 ^iuoamdo ;%sfldaKhl iudk mdoj,g iuqL fldaK iudk ksid&

OAC + OCA XOA 2 ^;%sfldaKhl mdohla la lsfuka

iEfok ndysr fldaKh wNHka;r

iuqL fldaK fofla tl;=jg iudk ksid&

by; idOkh l< m%fhh Nd;fhka fjk;a idOkh lsf .eg ^wkqfhh& idOkh lrk wdldrh oeka uid n,uq'

ksoiqk 2

O flakah jQ jD;a;h u; A, B iy C ,laIH msysgd we;' k AC = AB nj fmkajkak'

C

BA

O

idOkh:

2

2

2

313 wNHdih

o

C

BA

1' O flakah jq jD;a;hla u; A, B iy C ,laIH msysgd we;' k 30 nj fmkajkak'

2' P, Q, R iy S ,CIH jD;a;hla u; msysgd we;'

S

RQ

P nj idOkh lrkak'

170 171


3' O flakah jq jD;a;hla u; A, B" C iy D ,laIH msysgd we;'

B

D

CA

O

k nj fmkajkak'

O

Q

RP

4' P, Q iy R ,CIH O flakah jQ jD;a;fha mh u; msysgd we;' k nj fmkajkak'

^O yd Q hd lrkak'&

O

B

CA

5' O flakah jq jD;a;h u; A" B" iy C ,laIH msysgd we;' 2 ( ) nj fmkajkak'

31'4 jD;a;hl tlu LKavfha fldaK w;r inkaOh

BA

uyd jD;a;

LKavh

iq`M jD;a;

LKavh

jD;a;hla iy tu jD;a;hg w`osk ,o AB cHdh rEm igyfka oelaf' tu cHdh u.ska jD;a;h fmfoia follg fjka f' tla fmfoila jkafka cHdfhka iy uyd jD;a; pdmfhka jg jq uyd jD;a; LKavh hs'

wfkla fmfoi cHdfhka iy iq`M jD;a; pdmfhka jg jq iq`M jD;a; LKavh hs'

O

BA

XjD;a; LKavfha fldaKh

AB cHdfha fofl


P

X Y

QR

S

w`k ,o tlu LKAvfha fldaK fldaKudkhl wdOdrfhka ukskak' tla

tla fldaKfha Yd,;ajh mCId lrkak'

by; wdldrhgu jD;a;fha iq`M jD;a; LKAvhg wh;a jD;a; LKAvfha fldaK

lsysmhla we`o tu fldaK uek ,efnk w.hka mCId lrkak'

fuu l%shdldrlj, fhfuka tlu LKAvfha fldaK iudk nj y`kd.kakg we;' th m%fhla f,i my; oelaf'

m%fhh ( jD;a;hl tlu LKavfha fldaK iudk f'

tfia y`kd.;a m%fhh cHd;sl idOkhla u.ska ;yjqre lr .ksuq'

A

X Y

B

O

C

o;a;h: O flakah jq jD;a;fha XY cHdfha tlu me;af;a jD;a;h u; A" B iy C ,CIH msysgd we;'

idOkh l< hq;a;: ksudKh: OX iy OY hd lsu'

idOkh: jD;a;hl cHdhlska jD;a;fha flakah u; wdmd;kh lrk fldaKh mh u; wdmd;kh lrk fldaKh fuka fo.=Khla f'

by; oelajq m%fh .Kkh ls i|yd fhdod.kakd wdldrh uid n,uq'

rEmfha we;s f;dr;=re Wmfhda.S lrf.k fidhkak'

P

S

RQ30x

by; oelafjk rEmfha PQ = QR =PR iy f' w.h fidhuq' PQ = QR = PR ksid PQR ;%sfldaKh iumdo ;%sfldaKhls' iumdo ;%sfldaKhl wNHka;r fldaKhl w.h 600 ksid'

Q

^tlu jD;a; LKavfha fldaK iudk ksid&

;%sfldaKhl wNHka;r fldaKj, tl;=j 1800 ksid

( )30 60 +

ksoiqk 1

rEmfha we;s f;dr;=re fhdodf.k C fidhkak'

X100x

C

B

AD

174 175


XB = XA ksid XAB ;%sfldaKh iuoamdo ;%sfldaKhls'

' ''

^ iuoamdo ;%sfldaKhl iudk mdoj,g iuqL fldaK iudk f'&

ABX ;%sfldaKfha ^ ;%sfldaKhl fldaK 3 tl;=j 1800 ksid&

(

40

40

)

31'4 wNHdih

1' my; oelafjk wNHdij, x ys w.h fidhkak'1'

x

C

B

A

D

2'

x

PQ

RS

30x

40x

3'

x

AB = AC

50x

A

C

D

E

B

4'

x

110x

CB

A

D 5'

x

>

>

RQ

P

S

P

70x

6'

x

AD

CB

70x

7'

x120x

C

B

A

E

D

P

8' x

50x

E

D

CB

A 9'

xP

Q

30x70x

A

C

D B

60x

31'5 tlu jD;a; LKavfha fldaK iudk f' m%fh Nd; lrka wkqfh idOkh

ksoiqk 1

rEmfha we;s f;dr;=re weiqfrka AC = BD nj fmkajkak'

B

C>

>A

D

X

DC

31'5 wNHdih

1' AB$$CD k nj fmkajkak' 2' PX = QX k PQ$$SR nj fmkajkak'

B C

>>

A D

x

Q R

PS

x

176 177


3' AD // BC k" AX = DX nj fmkajkak' 4' nj fmkajkak'

B

C

AD

X

>

>

B C

A

X

Y

5'

fmkajkak'

6'

E

A

B

C

D

RP

Q

B

A

D

X

C

7' XB = XC k AC = BD 8' XS = XR k" XP = XQ nj fmkajkak' nj fmkajkak'

X

B

C

D

A

X

S

R

Q

P

31'6 wO jD;a;fha fldaK

jD;a;fhka y wvla jq jD;a; pdmh wO jD;a;hla f,i ye`kaf'

A B

wO jD;a;fha fldaKh

wO jD;a;h

flakah

jD;a;fha flakah yryd fLdjla wefuka jD;a;h wO jD;a; follg fjkaf'

wO jD;a;h u; ,laIHhla wO jD;a;fha fofl,jrg hd lsfuka iEfok fldaKhg wO jD;a;fha fldaKh hehs lshkq ,ef'

wO jD;a;fha fldaKj, ,laIK y`kd .ekSu iòyd my; oelafjk l%shdldrlfuys fhfokak'

ls%hdldrlu 1

lvodishla u; ljlgqj Nd;d lrka jD;a;hla weòskak' tu jD;a;fha flakah yryd ir, fLdjla w`kak' tg jD;a;h" wO jD;a; follg fjka f'

tla wO jD;a;hla u; ,laIHhla ,l=Kq lrkak' tu ,CIHh wO jD;a;fha fofl,jrg hd lrkak' tg jD;a; pdmh u; wO jD;a;fha fldaKhla ,ef'

fldaKudkh Nd;fhka wO jD;a;fha fldaKh ukskak'

wO jD;a;fha fldaKh 90x nj Tng fmfkkq we;' fuf,iu ;j;a jD;a; lsysmhla we| tAjdfha wO jD;a;fha fldaK we| w.h ukskak' by; iòyka l%shdldrlfuys wO jD;a;fha fldaKh yeu gu iDcqfldaKhla jk nj Tng ksCIKh lsug yelsjkq we;'

by; oelafjk inkaOh cHd;sl idOkhla u.ska ;yjqre lr .ksuq'

178 179


OA B

Y

X

'

o;a;h : O flakah jq jD;a;fha AB YalNhla jk w;r X yd Y hkq rEmfha olajd we;s mrsos jD;a;h u; msys ,laIH f'

idOkh l< hq;a;: iDcqfldaKhla nj' idOkh: hkq AYB pdmh u.ska flakah u; wdmd;kh lrk fldaKh

f'

AYB wO jD;a;hla ksid AOB IalNhla f' = iDcqfldaK 2 AYB pdmh u.ska jD;a;fha b;s fldgi u; wdmd;kh lrk fldaKh f'jD;a; pdmhla u.ska flakah u; wdmd;kh lrk fldaKh" tu pdmfhka jD;a;fha b;s fldgi u; wdmd;kh lrk fldaKh fuka fo.=Khla jk ksid"

iy ksid 2 } iDcqfldaK 2

iDcqfldaK 1

by; idOkh u.ska ;yjqre l< inkaOh m%fhla f,i my; oelaf'

m%fhh: wO jD;a;fha fldaKh iDcqfldaKhla f'

my; oelafjk ksoiqka u.ska by; m%fhh weiqfrka .Kkh ls lrk wdldrh y`kd .ksuq'

rEm igyfka O flakah jk jD;a;fha olajd we;s o;a; weiqfrka ACD

>

w.h fidhuq'

o

C

BA

D

50x

tlu jD;a;fha wr neka

DO = OB ;%sfldaKhl iudk mdoj,g iuqL fldaK iudk ksid

ksoiqk 1

O

R

QP

S

20x we;s jD;a;fha PQ hkq IalNhls' QPR

>

= 20x k yd PS = QR k RPS

>

w.h fidhkak'

20 90

180 181


316 wNHdih

1' my; oelafjk tla tla jD;a;fha flakah O j,ska oelaf' we;s o;a; wkqj x ys w.h fidhkak'

(i)

Ox

C

BA30x

(ii)

Ox

(iii)

C

B

A 60x

Ox

D

(iv)

O

x

C

BA

D

50x

(v)

Ox C

B

A

D

110x

31'7 wO jD;a;fha fldaKh iDcqfldaKhla f hk m%fhh Nd;fhka wkqfhka idOkh lsu

ksoiqk 1

PQ" PQRS jD;a;fha IalNhls' RS" X olajd .= lr we;' + nj idOkh lrkak'

idOkh(-

P

RS

Q

X

182 183


ksoiqk 2

A B C D

E

X Y

we;s rEmfha jD;a; foflys flaka X iy Y f' nj fmkajkak'idOkh ( AC, X yryd hk neka AC" X flakah jq jD;a;fha IalNhls'

317 wNHdih

1' rEmfha oelafjk ABCD jD;a;fha IalNh AC f'

C

B

A

D

nj fmkajkak'

2' rEmfha oelafjk PQRS jD;a;fha IalNh PR f' PQ = RS R

Q

P

S

k PQRS iDcqfldaKdi%hla nj fmkajkak'

3' PQ hkq PQR jD;a;fha IalNhls' PQ = QX iyR

Q

P

X

PR =QR= RX' x nj fmkajkak'

4' AC hkq ABCD jD;a;fha IalNhls' BC$$AD f'

D

C

B

>

>

A

ABCD iDcqfldaKdi%hla nj fmkajkak'

5' Yd, jD;a;fha IalNh PSQ o" l=vd jD;a;fha

Q

X

SP

R

IalNh SQ f' RQ" X ,CIHfha l=vd jD;a;h lefma' nj fmkajkak'

184 185


6' PR hkq we; jD;a;fha IAlNhls'

S

P

Q

R nj fmkajkak'

7' jD;a; fol P yd Q ,CIHj, fokh f' jD;a; P

Q XY

fofla IAlN PX iy PY f' XQY ir, fLdjla nj fmkajkak'

Y% wNHdih

we;s o;a; rEmigykaj, ,l=Kq lr we;s .eg i`okak'

CB

o

A

1' ABC jD;a;fha flakah O f' w.h

fidhkak'

2' ABCD jD;a;fha IAlNh BD f'

CB

D

A

BC = CD iy w.h fidhkak'

3' we;s ABCD jD;a;fha flakah O f' BC//OD o

C

B

>>

o

D

A

o f' fldaKfha w.h fidhkak'

C

B

o

X

A

4' ABC jD;a;fha flakah O f' BC = CX jk fia AC" X olajd .= lr we;' nj fmkajkak'

5' ABCD ,CIH jD;a;h u; msysgd we;' AD = BC f'

C

BA

D

DB = CA nj fmkajkak'

6' PQ hkq O flakah jq jD;a;fha IalNhls' QR//OS f'

R

QP o

S

SR = SP nj fmkajkak'

186 187

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Grade 10 Maths 13