TRAN DUC HUYEN - NGUY�N DUY HIEU

TAI LIEU CHUYEN TOAN . �-

TR UNG HOC PHO THONG

nJ,,,,,�n; � ,•(A�rV A

HINH HOC KHONG GIAN · •

NHA XUAT BAN GIAO DVC Vll;T NAM

    

{[[W+bz0FkV43GmRt7u4DpvuYxd]]}

. . ,

I

I

I

I

I

I

I

I

I

I

I

I I

I I

I

I

I

I

I

I

I

I

I

I

I

I

LOINOIBAU

�P theo 8¢ sach Gidi todn danh cho h9c sinh lop chuyen, nhom tac gia truong THPT chuyen Le Hong Phong bien so�n 89 sach Tai lifu chuyen toanTHPT theo chuyen de nh�m giup cac em h9c sinh tV h9c de nam duqc tr9ng tam kien thuc va luy�n t�p giai toan, qua do giup h9c sinh tlf on thi vao dc;ti h9c d�t ket qua tot.

Cuon sdch Tai lifu chuyen Toan THPT chuyen de: Hinh h9c khong giangom 2 phan chfnh:

Phan m¢t: Lf THUYgT VA PHlf ONG PHAP GIAI TOAN

Gorn 8 chuang Chuang l. HINH LANG TRI) Chuang 2. HINH HOP Chuang 3. HINH CHOP Chuang 4. HINH cAu Chuang 5. HlNH TRl) Chuang 6. HlNH NON Chuang 7. cAc BAI TOAN VE KHOANG CACH Chuang 8. cAc BAI TOAN VE GO(

Phan hai: UNG Dl)NG OE GIAI CAC OE THI TUYEN SINH DAI HQC

Phan nay gom m¢t s6 de thi tuyen sinh d�i h9c nhung nam gan day, m¢t so de thi mau ch9n l9c va nhung bai toan hlnh h9c khong gian thuong g�p trong giai toan.

Rat mong cu6n sach nay giup fch cho cac em h9c sinh ttf h9c va tV on thi vao d�i hQC thanh cong.

M�c du rat c6 gang trong qua trlnh bien so�n nhung nhung khiem khuyet la kh6ng the tranh khoi, chung t6i rat mong nh�n duqc cac y kien dong gop cua cac quy thay c6 va cac em hQC sinh yeu toan de cac Ian tai ban sau, cuon sach duc;,c hoan chinh hon.

MQi y kien dong gop xin gui ve theo dja chi: Ban Bien t�p Toan - Tin, Cong ty CP Ojch v1,1 xuat ban giao d1,1c Gia f)jnh - Nha

xuat ban Giao d1,JC Vi�t Nam - 231 Nguyen Van Cu, qu�n 5, TP. Ho Chf Minh.

NHOM TAC GIA

3

    

{[[W+bz0FkV43GmRt7u4DpvuYxd]]}

/, ... , TBUYIT , ., . , . , .· VA PBDdNG PBAP GIAI TOAN

HUONG 1. HlNH LANG TR\J ········································-························

A. TOM TAT Lt THUYfr

I. DJNH NGHiA

Cho hai m�t philng (a) va ·(a')song song vm nhau. Tren (a) cho da giac 16i A

1A

2" . .A

n. Qua cac dinh

Ai, A

2 , ••• ,An

ta ve cac ducmg thilng song song vai nhau va cit mp (a') lful lm;rt t�i A 'i, A '

2 , ••• ,A 'n.

Ilinh g6m hai da giac A1A

2 •• .A

n va

A 'i A '2

••• A 'n

va cac hinh binh hanh A

1A\A'

2A

2, A

2A'

2A\A

3, • • • ,

An A '

n A 'i A

1 dugc gQi la hinh Zang

trlj. va dugc ki hi�u la A,A

2···A

n . A\ A'

2···A'

n.

Hai da giac A1 A

2 .. .A

n va A 'i A '

2 ... A '

n gQi la hai mJt day, cac hinh binh hanh

A1A 'i A '

2 A2 , A

2A 12 A\ A3 , • • • , An

A 'n A 'i A, dugc gQi la cac mJt hen, cac c�

khong thUQC hai day d�U song song Vffi nhau Va dugc gQi la C� ben. Khoang each gifra hai day gQi la chi�u cao cua lang tr\1.

II. TINH CHAT

- Hai day la hai da giac bing ·nhau, c6 c�nh doi mot song song va bing nhau.- Cac c�nh hen song song va bing nhau.- Cac mJt ben la hinh binh hanh.

5

III. PHAN LO�

- ffinh ling tr(l 4,mg la hinh ling tl'\l c6 C\illl} hen vuong g6c vai �t day.- mnh lllng tr(l xien la hinh lang trv c6 c� hen khong vuong g6c vai �t day.- mnh /ling trv, tliu la hinh lang trv. dung c6 day la da giac d�u.

Lang t11,1 dung Lang tn,J xi�n

.

IV. C6NG THUC

6

- Th� tich cua kh6i lang trv bfulgtich st, cua di�n tich day va chi�ucao cua kh6i ling trv d6

V = B.h- Di� tich xung quanh bfulg t6ng

di�n tich cac �t ben8xq = L8m*t ben

- Di�n tich toan phAn bfulg t6ng cuadi�n tich xung quanh va di�n tichhai day

Stp

=S +2B. xq

B. cAc BAI TO.AN VA PHUONG PHAI> GIAI

i Tf nh the tfch khoi lang tr\J

I. PHU'ONG PHA.PD� tinh th� tich cac kh6i Hing tf\l ta th\lC hi�n cac buoc- Xac djnh chan ducmg cao- Ve ducmg cao- Tinh ducmg cao h- Tinh di�n tich day B- Tirih th� tich lang tf\l theo cong thuc V = Bh.

II. cAc BAI TOAN

(}Jai todn 1. Tinh th� tich kh6i lang tf\l l\lC giac d�u c6 Cl;lnh day bing a vaducmg cao 3a.

3a I

I

)--------- .... ., '

,, ', ., '

,,. '

Ta c6 chi€u cao lang tf\l la: h = 3a.

D·� ' h d' . l� I' B 6 a2

.fs 1�n tic ay cua ang tr\l a: = .-4- ·.

Th ;: 'h · 1� l' V ·Bh 3. 6 a2

-J3 9a3

.Jse tic cua ang tf\l a : = . = a. . -4- =

2 .

7

8

cBtli todn 2. Cho lang tn.i tam giac d�u ABC.A'B'C' c6 c�nh day bing a,

khoang each tir tam O cua tam giac ABC d�n mijt phing (A'BC) bing i. Tinh th€ tich cua kh6i Hing tf\l da cho.

fiiai C'

G9i I la trung di€m cua BC, 0 la tam cua tam

giac ABC nen O thuoc AI va OI = _!_ AI . . 3 D\1'11g OH .LA'I �i H, ta·c6:

{BC .l AI => BC .l (A I AI) => BC l OH . BC .l AA'

L�i c6 OH.LA'! nen OH .L( A'BC). . . a V�y OH= d(O, (A'BC)) =

6.

Dl\t AA'= x (x > 0), xet tam giac vuong A' AI ta c6:

A 'I= .JAA '2 + AI2

= x2 + (a.Ja)2 = Jx2 + 3a2

•

2 4

Hai tam giac vuong OHi va A' AI d6ng d�ng, suy ra OH

=

OI ¢:) OH.A I I = QI.AA I

AA' A'I

<::) a,.Jaa2

+ x2 = a.fa .x (vi OI =!AI= a.fa)6 4 6 3 6

<::)•� = Ja.x <::) 3:2

+ x2 = 3x2

2 3a2

. a./6 ¢:) x =-<::) x =--.

·8 4 Toe tich cua kh6i ling tI'\l ABC.A'B'C'

V = S .AA'= a

2 .Ja .a .Ja ;,,, a..f2aa

ABC 4 4 16

B

'914911 I Tfnh di¢n tfch h1nh lang trl,J

I. PHUONG PHAPCac m�t ben cua hinh Hing tr\l la cac hinh binh hanh. Hai day cua hinh lang tr\lla hai da giac bing nhau. 08 tinh di�n tich hinh lang tr\l ta thgc hi�n cac bu&c :- Xac dinh hinh tinh cac m�t- Tinh di�n tich tirng m�t- Tinh di�n tich la.rig tr\l theo cac cong thuca) Di�n tich xung quanh bing t6ng di�n tich cac m�t ben

s = ""s • bA xq LJ ml,lt enb) Di�n tich toan phfui bkg tf>ng cua di�n tich xung quanh va di�n tich hai day

S =S +28.tp · xq

II. cAc BAI TOAN

<.Bai todn 3. Tinh .di�n tich xung quanh va di�n tich ·toan phfui cua hinh lang tf\ll\lC giac d€u CO c?nh day bkg a Va duong cao 3a.

<Jiai

Ta c6 chi€u cao lang tf\l la : h = 3a (xem hinh bai toan 1 ).

D.

A t' h d' . ' Ix- I' B 6 a 2

../3 3a2

../3 t�n IC ay cua aug 1f\l a : = .-4- =

2 .

M6i mJt hen cua lang tf\l la hinh chfr nhjt c6 <li�n tich S = 3a2

=> S = 6.3a2 = 18a2

xq

2 3a2../3 2 � => S1

= S +2B = 18a +2 = 3a (6+-v3).p xq 2 .

(Bai to4n 4. Cho ling 1f\l tam giac d€u ABC.A'B'C' 'co c?nh day bing a,

khoang each tu tam o cua tam giac ABC� m�t phing (A'BC) bkg �.

Tinh di�n. tich toan phfui cua hinh lang tf\l.

9

'

A

B

' ' ' ' '

I

c·

Th h, 'nh' b'' ' 2 d- ' h'� ' 1� l' AA' af6 eo c ung m1 a ai toan , ta a co c 1eu cao cua ang tfl,l a : = 4 .

C' � b� ' 1� l' ' h' h h- h� ' k' h h ' - ' af6 ac m<;tt en cua ang trt) a cac m c u n <;tt co 1c t uac a va 4

.

Hai day la hai tam giac d€u c6 C<;tnh bing a. V�y di�n tich toan phfi.n cua Hing tf\l ABC.A'B'C' la:

Stp = Sxq + 2Soay = 3s ABB'A' + 2S ABC

= 3AA' .AB+ a2 J3

= 3a2 II+ a2

J3=

a2 J3(3Ji + 2).vs 2 4

lft'n � chuae 3

10

I Phan lo�i hlnh lang trt,1 �heo dvong cao

D� giai cac bai toan v€ hinh Hing tf\l, di€u quan tr9ng nhftt la xac dinh duqc ducmg cao. Ngucri ta phan lo<;ti Hing tf\l theo cac ki�u ducrng cao nhu sau: Lo{li 1 : DU'(1NG CAO LA C�NH BEN (Tftt ca cac lang tr\l d€u thi c6 ducrng cao la C<;tnh ben).

<Bai todn J. Cho Hing tr\l tam giac d�u ABC.A'B'C' c6 qmh day b�ng a va qmh hen b�ng a.Jj. Tinh th€ tich va di?n tich toan phfrn cua Hing tr\l.

qitii A'

B'

a.fa

B Do ABC.A'B'C' la lang tf\l tam giac d�u nen day ABC la tam giac d�u va duang cao la qmh ben: h = AA'= a.Jj. ·

D·� ' h d' B s· a2 .Jj u;n tic ay = ABC = -4-

.

Th� tich V = B.h = SABC.AA' = a2.Jj .a.Jj = 3a3

.4 4

D·� ' h ' h). S S 2B 3 2 r:;3 2 a2 .Jj ?a2 .fj 1entic toanp an:

1 = + = a v.:t+ .--= .

P xq 4 2

<Bai todn 6. Cho lang tf\l dung ABC.A'B'C' c6 day la tam giac ABC vuong t�i B. AB = a, BC = a.Jj, AA'= 2a. Tinh th� tich va di?n tich toan phfrn cua lang trl,l. B'

qitii

'no ABC.A'B'C' la lang tf\l dung nen duang cao la c�nh ben : h = AA' = 2a.

D·� 'hd' B S 1ABBC a2

Jj u;n tic ay. = ABC =2 . =-2-

.

Th� tich V = B.h = S ABC .AA' = a 2

Jj .2a = a 3 Jj. 2

Tam gi�c ABC vuong t�i B nen ta c6 : AC= �BA 2 + BC2 = 2a .

C'

c

11

12

.2 ,fj

Dien tich toan ph.ln : S = S + 2B = 2a(a + 2a + aF3) + 2.� . tp xq2

= 2a2 (3 + ,fj) + a2 ,fj = 3a2 (2 + ,fj ).

Lo(li 2: DlfONG CAO NAM BEN TRONG LANG TRT)

<.Bai todn 7. Cho lang tf\l tam giac ABC.A'B'C' c6 day la tam giac d�u qmh b�ng a, A' A = A'B = A'C = aF3. a) Chung minh tu giac BB'C'C la hinh chu nh�t.b) Tinh th� tich cua lang tn,1.

<;itli A' c·

B a) G9i G la trc;mg tam tam giac d�u ABC va g9i M la trung di�m BC, ta c6:A' A =A;B = A'C, suy ra A'G vuong g6c vcri mp(ABC).

BC ..L A I G} => BC ..L AA'. BC..LAG

Ma.BB' II AA':::>BC..LBB'. Tu giac BB'C'C la hinh binh hanh va c6 m('>t g6c vuong nen la hinh chu nh�t.

b) ABC 1, ·' d}. A AM aF3AG. aF3 a tam giac eunen: =--; =--. . 2 3

' I 2 2 __ �3a

2 _ 3a2

_ _ �24a2

__ 2a-.J6.Chieu cao Hing tf\l : h = A I G = "AA I - AG 9 9 3

D ·A 'hd' B S a2 F3 1�n tic ay = ABC =-4-.

    

{[[W+bz0FkV43GmRt7u4DpvuYxd]]}

Th� tich Jang tn,1 : V = B.h = S ABCA' G = a2 f3. 2a[6 = a

3 -/2- .

c:Bai toan 8. Cho Jang tr\l ABCD.A'B'C'D' c6 day la hinh thoi c.,mh bing a.G6c BAD= 60° , A' A = A'B = A'D = a. Tinh th� tich cua Jang tf\l.

qitli D' C'

Tam giac ABD can c6 mc)t g6cbing 60° nen la tam giac d�u. G9iG la tr9ng tam tam giac d�u ABDva g9i O la trung di�m BO, ta c6A'A = A'B = A'D suy ra A'Gvu6ng g6c v6i mp(ABCD). ABD la tam giac d�u nen :

AO= aJ3 ·AG= aJJ.2 ' 3

Chi�u cao Jang tr\l :A B

h = A 'G = )A'A2 -AG2 = Ja2 - 3�2

= J6�2

= •:.

D·� ' h d' S 2S a2J3

I�n tic ay ABCD = ABD =

-2-.

Th�' hi- V B h S A'G a2J3

a./6 a3 J2 etic angtr\l: = . = ABCD: =-2-.-

3-=-

2-.

Lo9i 3 : DUONG CAO NAM TR ONG MQT M� T BEN

c:Bai toan 9. Cho Jang tf\l tam giacABC.A'B'C' c6 day ABC la tam giacvu6ng can t�i B, AB = a. M�t benAA'C'C la hinh thoi va vu6ng g6c v6im�t day. C�nh ben AA' hQ'P v6i m�t daymc)t g6c 60°. Tinh th� tich cua Jang tf\l.

(jiai

G9i H la hinh chi�u vu6ng g6c cua A'xu6ng qnh AC. Do m�t ben AA'C'Cvu6ng g6c v6i m�t day nen ta c6 A 'Hvu6ng g6c v6i mp (ABC), suy raA'AH=60° .

A'

B

C'

13

14

Tam giac �' AC can c6 m(>t g6c b�ng 60° nen la tam giac d�u . . � � ACJ3 ahJ?, a/6 Ch1eu cao lang tru : h = A' H = = = -- .. . 2 2 2

Di�n tich day B = S ABC = a2

Th� 'h 1� V Bh S \'H a 2 a/6 a3 /6 e tic ang trµ: = . = ABC' = 2.-2- = -4-.

(}Jai todn 10. Cho Jang trµ ABC.A'B'C' c6 day la tam giac ABC vuong t1;1i B, AB = a, BC = aF3, AA'= 2a. Hinh chi�u vuong g6c cua A' xu6ng m�t day trung v6'i trung di�m H ctia c1;1nh·AC. Tinh th� tich cua lang trµ.

A'

B

Ta c6 AC = � BC2 + BA 2 = 2a .

qidi

c·

Ta c6 H la hinh chi�u vuong g6c cua A' xu6ng m�t day nen ta c6 A'H vuong g6c v6'i mp (ABC), suy ra chi�u cao lang tn,i la :

h = A'H =�A'A2 -AH2

= �4a2 -a2 = aJ3.

D · A ' h d' B S a2 J?, 1�11 tic ay = ABC = -2-.

2,fj 3 Th� tfch lang trµ: V = B.h = S ABC.A 'H = a

2 3 .aJ3 = 3;

{ff!r7":"' HUONG 2. HiNH HOP·�

:-·:_ .

1/: ),c __ ._;_

,.'.

.•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

A. T6M TAT Lt THUYtr

I. DJNH NGHiA

Hinh h9p la hinh Hing ttv co day la hinh binh hanh. K.hoang each gifra hai day gQi la chi�u cao cua hinh h(>p.

II. TINH CHAT- Ca 6 m�t cua hinh h9p dBu la hinh binh hanh.- Hinh h9p la hinh Hing ttv ma mQi m�t dBu co th� l§y lam m�t day.- Bo� n6i hai dinh khong thu(k m(>t m�t gQi la ducmg cheo cua hinh h(>p.- Ba ducmg cheo cua hinh h(>p d6ng quy �i m(>t di�m, di�m do la tam dfii

xung cua hinh h(>p.

III. PHAN LO�

ffinh h{,p dU'llg la hinh h(>p co c�nh ben vuong goc vcn day.ffinh h{,p chir nhit la hinh h(>p dung co day la hinh chu nh�t.ffinh lip phuong la hinh h(>p co tAt ca cac m�t la hinh vuong.• Hinh h9p chfr nh�t c6 ba kich thu6c a, b, c

thi ducmg cheo co d(> dai

d= �a2 +b2 +c2

• Hinh l�p phuang co c� a thi d = al3 .

A' D'

. ��------- ----- D,/

,. ,.

,.

B c

15

IV. CONG THUC

- Th� tich cua kh6i h(>p duqc tinh theo th� tich cua kh6i Hing tr\l

V=Bh

- Di;tc bi�t th€ tich cua kh6i h(>p chu nh�t bing tich cua ba kich thu6c

V =a.b.c va th€ tich cua kh6i l�p phuong bing l�p phucmg s6 do cua qnh

V =a3

- Di�n tich xung quanh bing t6ng di�n tich cac mi;lt ben

s = "s . �A

xq L.. mi.i.t,V<On

- Di�n tich toan phAn bing t6ng cua di�n tich xung quanh va di�n tich hai day

S =S +2B. tp xq

B. cAc BAI TOAN VA PHUONG PHAP GIAI

I Tf nh the tfch khoi h<)p

I. PHUONG PHA.P

D� tinh th€ tich cac kh6i h(>p ta th\fc hi�n cac bu6c

- Xac djnh chan dm'mg cao

- Ve ducmg cao

- Tinh ducmg cao h

- Tinh di�n tich day B

- Tinh th� tich khf>i h9p theo cong thuc V = B.h

II. cAc BAI TOAN

-, , <.Bai todn 1. Cho hinh h(>p dung ABCD.A'B'C'D' c6 day ABCD la hinh thoi

va BAD = 60° . Ducmg cheo BD' hqp v6i day m(>t g6c 60°. G<;>i E la trungdi�m cua AD. Mi;lt ph�ng BED' c�t hinh h(>p theo m(>t thi�t di�n c6 di�n tichbing 4 F9 cm2

. Tinh th€ tich cua khf>i h(>p da cho.

16