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1. √Ÿª‡√¢“§≥‘µ∑’˧≈⓬°—π
§«“¡§≈⓬4
√Ÿª‡√¢“§≥‘µ Õß√Ÿª‡ªìπ√Ÿª∑’˧≈⓬°—π ‡¡◊ËÕ√Ÿª‡√¢“§≥‘µ∑—Èß Õß¡’√Ÿª√à“߇À¡◊Õπ°—π ¢π“¥
Õ“®‡∑à“°—πÀ√◊Õ·µ°µà“ß°—π°Á‰¥â
°‘®°√√¡∑’Ë 1
√Ÿª‡√¢“§≥‘µµàÕ‰ªπ’È ‡ªìπ√Ÿª∑’˧≈⓬°—πÀ√◊Õ‰¡à
1.
√Ÿª ’ˇÀ≈’ˬ¡ Õß√Ÿªπ’È ................................................................. (§≈⓬°—π / ‰¡à§≈⓬°—π)
2.
√ŸªÀⓇÀ≈’ˬ¡ Õß√Ÿªπ’È .............................................................. (§≈⓬°—π / ‰¡à§≈⓬°—π)
98 ·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
3.
√ŸªÀⓇÀ≈’ˬ¡ Õß√Ÿªπ’È .................................................................‹ (§≈⓬°—π / ‰¡à§≈⓬°—π)
4.
√ŸªÀⓇÀ≈’ˬ¡ Õß√Ÿªπ’È .................................................................. (§≈⓬°—π / ‰¡à§≈⓬°—π)
5.
√ŸªÀ°‡À≈’ˬ¡ Õß√Ÿªπ’È ................................................................. ‹(§≈⓬°—π / ‰¡à§≈⓬°—π)
6.
√Ÿª ’ˇÀ≈’ˬ¡ Õß√Ÿªπ’È ............................................................ (§≈⓬°—π / ‰¡à§≈⓬°—π)
99·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
2. √ŸªÀ≈“¬‡À≈’ˬ¡ Õß√Ÿª∑’˧≈⓬°—π
∫∑𑬓¡
√ŸªÀ≈“¬‡À≈’ˬ¡ Õß√Ÿª§≈⓬°—π °ÁµàÕ‡¡◊ËÕ √ŸªÀ≈“¬‡À≈’ˬ¡ Õß√Ÿªπ—Èπ ¡’
1. ¢π“¥¢Õß¡ÿ¡‡∑à“°—π‡ªìπ§ŸàÊ ∑ÿ°§Ÿà
·≈– 2. Õ—µ√“ à«π¢Õߧ«“¡¬“«¢Õߥâ“π§Ÿà∑’Ë ¡π—¬°—π∑ÿ°§Ÿà‡ªìπÕ—µ√“ à«π∑’ˇ∑à“°—π
µ—«Õ¬à“ß
®“°√Ÿª °”Àπ¥„Àâ√Ÿª ABCDE √Ÿª JKLMN ®–‰¥â«à“
1. A = J, B = K, C = L, D = M ·≈– E = N
2. ABJK
= BCKL
= CDLM
= DEMN
= EANJ
°‘®°√√¡∑’Ë 2
1. ®ß‡¢’¬π Àπâ“¢âÕ§«“¡∑’Ë∂Ÿ°µâÕß ·≈–‡¢’¬π Àπâ“¢âÕ§«“¡∑’˺‘¥
................. (1) √ŸªÀ≈“¬‡À≈’ˬ¡∑’˧≈⓬°—π ®–¡’¡ÿ¡‡∑à“°—π¡ÿ¡µàÕ¡ÿ¡ ·µàÕ—µ√“ à«π¢Õߥâ“π§Ÿà
∑’Ë ¡π—¬°—π‰¡à®”‡ªìπµâÕ߇∑à“°—π
................. (2) √ŸªÀ≈“¬‡À≈’ˬ¡∑’˧≈⓬°—π ®–¡’Õ—µ√“ à«π¢Õߥâ“π§Ÿà∑’Ë ¡π—¬°—π‡∑à“°—π ·µà
‰¡à®”‡ªìπµâÕß¡’¡ÿ¡‡∑à“°—π¡ÿ¡µàÕ¡ÿ¡
................. (3) √Ÿª ’ˇÀ≈’ˬ¡®—µÿ√— Õß√Ÿª„¥Ê ‡ªìπ√Ÿª∑’˧≈⓬°—π
M
L
KJ
N
A B
CE
D
100 ·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
................. (4) √Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“ Õß√Ÿª„¥Ê ‡ªìπ√Ÿª∑’˧≈⓬°—π
................. (5) √Ÿª ’ˇÀ≈’ˬ¡¢π¡‡ªï¬°ªŸπ Õß√Ÿª„¥Ê ‡ªìπ√Ÿª∑’˧≈⓬°—π
................. (6) √Ÿª “¡‡À≈’ˬ¡Àπâ“®—Ë« Õß√Ÿª„¥Ê ‡ªìπ√Ÿª∑’˧≈⓬°—π
................. (7) √Ÿª “¡‡À≈’ˬ¡¥â“π‡∑à“ Õß√Ÿª„¥Ê ‡ªìπ√Ÿª∑’˧≈⓬°—π
................. (8) √Ÿª “¡‡À≈’ˬ¡Àπâ“®—Ë« Õß√Ÿª„¥Ê ∑’Ë¡’¡ÿ¡¬Õ¥‡ªìπ¡ÿ¡©“° ‡ªìπ√Ÿª∑’˧≈⓬°—π
................. (9) √Ÿª ’ˇÀ≈’ˬ¡√Ÿª«à“« Õß√Ÿª„¥Ê ‡ªìπ√Ÿª∑’˧≈⓬°—π
................. (10) √Ÿª “¡‡À≈’ˬ¡∑’Ë¡’¥â“π‡∑à“°—∫ 10, 10 ·≈– 4 Àπ૬ ·≈–√Ÿª “¡‡À≈’ˬ¡∑’Ë¡’
¥â“π‡∑à“°—∫ 5, 5 ·≈– 3 Àπ૬ ‡ªìπ√Ÿª∑’˧≈⓬°—π
2. ®“°√Ÿª °”Àπ¥„Àâ√Ÿª ’ˇÀ≈’ˬ¡ ABCD ·≈–√Ÿª ’ˇÀ≈’ˬ¡ PQRS ‡ªìπ√Ÿª ’ˇÀ≈’ˬ¡∑’˧≈⓬
°—π ®ßÀ“
(1) Õ—µ√“ à«π¢Õߧ«“¡¬“«¢Õߥâ“π§Ÿà∑’Ë ¡π—¬°—π
(2) §«“¡¬“«¢Õߥâ“π∑’ˇÀ≈◊Õ
(3) §«“¡¬“«√Õ∫√Ÿª¢Õß√Ÿª ’ˇÀ≈’ˬ¡∑—Èß Õß√Ÿª
(4) Õ—µ√“ à«π¢Õߧ«“¡¬“«√Õ∫√Ÿª¢Õß√Ÿª ’ˇÀ≈’ˬ¡∑—Èß Õß√Ÿª
«‘∏’∑” ‡π◊ËÕß®“° ABCD PQRS ®–‰¥â
(1) Õ—µ√“ à«π¢Õߧ«“¡¬“«¢Õߥâ“π§Ÿà∑’Ë ¡π—¬°—π¢Õß ABCD ·≈– PQRS §◊ÕABPQ
= CDRS
= DASP
= BCQR
AB5
= ........................................................
(2) ®“°¢âÕ (1)
À“§«“¡¬“«¢Õߥâ“π AB ®–‰¥âAB5
= ..........................................
AB = ............................... Àπ૬
S
P
C
5.6
D
6.3
BA
4.9
Q5
R
3.5
101·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
À“§«“¡¬“«¢Õߥâ“π RS ®–‰¥â
....................................... = .....................................
....................................... = .....................................
RS = .....................................
À“§«“¡¬“«¢Õߥâ“π SP ®–‰¥â
....................................... = .....................................
....................................... = .....................................
SP = .......................... Àπ૬
(3) §«“¡¬“«√Õ∫√Ÿª¢Õß ABCD ‡∑à“°—∫ ..............................................................Àπ૬
= .................................................................Àπ૬
§«“¡¬“«√Õ∫√Ÿª¢Õß PQRS ‡∑à“°—∫ ................................................................Àπ૬
= .................................................................Àπ૬
(4) Õ—µ√“ à«π¢Õߧ«“¡¬“«√Õ∫√Ÿª¢Õß ABCD µàÕ§«“¡¬“«√Õ∫√Ÿª¢Õß PQRS
‡ªìπ ....................................................................................................................................
¢âÕ —߇°µ
Õ—µ√“ à«π¢Õߧ«“¡¬“«√Õ∫√Ÿª¢Õß√ŸªÀ≈“¬‡À≈’ˬ¡∑’˧≈⓬°—π ®–‡∑à“°—∫
Õ—µ√“ à«π¢Õߥâ“π§Ÿà∑’Ë ¡π—¬°—π
3. ®“°√Ÿª °”Àπ¥„Àâ√Ÿª ’ˇÀ≈’ˬ¡§“ßÀ¡Ÿ ABCD §≈⓬°—∫√Ÿª ’ˇÀ≈’ˬ¡§“ßÀ¡Ÿ WXYZ
®ßÀ“
(1) Õ—µ√“ à«π¢Õߧ«“¡¬“«¢Õߥâ“π§Ÿà∑’Ë ¡π—¬°—π
(2) æ◊Èπ∑’Ë¢Õß√Ÿª ’ˇÀ≈’ˬ¡∑—Èß Õß√Ÿª
(3) Õ—µ√“ à«π¢Õßæ◊Èπ∑’Ë¢Õß√Ÿª ’ˇÀ≈’ˬ¡∑—Èß Õß√Ÿª
D C
BA
4
6 W X
10.5
Y
9
15Z
102 ·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
«‘∏’∑” ‡π◊ËÕß®“° ABCD WXYZ ®–‰¥â
(1) Õ—µ√“ à«π¢Õߧ«“¡¬“«¢Õߥâ“π§Ÿà∑’Ë ¡π—¬°—π¢Õß ABCD ·≈– WXYZ §◊ÕBCXY
= ....................... = ....................... = .......................
BC10 5.
= ....................... = ....................... = .......................
(2) ®“°¢âÕ (1)
À“§«“¡¬“«¥â“π CD ®–‰¥â
....................................... = .....................................
CD = ......................... Àπ૬
À“§«“¡¬“«¥â“π ZW ®–‰¥â
................................................ = ...........................................................................
................................................ = ...........................................................................
ZW = .................................................................Àπ૬
Ÿµ√ æ◊Èπ∑’Ë√Ÿª ’ˇÀ≈’ˬ¡§“ßÀ¡Ÿ = 12
º≈∫«°¢Õߥâ“π§Ÿà¢π“π Ÿß
æ◊Èπ∑’Ë√Ÿª ’ˇÀ≈’ˬ¡§“ßÀ¡Ÿ ABCD = .................................................................
æ◊Èπ∑’Ë√Ÿª ’ˇÀ≈’ˬ¡§“ßÀ¡Ÿ WXYZ = .................................................................
æ◊Èπ∑’Ë√Ÿª ’ˇÀ≈’ˬ¡ ABCD= ...................................................................
æ◊Èπ∑’Ë√Ÿª ’ˇÀ≈’ˬ¡ WXYZ
= .................................................................
®–‰¥âÕ—µ√“ à«π¢Õßæ◊Èπ∑’Ë¢Õß√Ÿª ’ˇÀ≈’ˬ¡ ABCD µàÕæ◊Èπ∑’Ë¢Õß√Ÿª ’ˇÀ≈’ˬ¡ WXYZ
෈ҡѺ .................................
¢âÕ —߇°µ
Õ—µ√“ à«π¢Õßæ◊Èπ∑’Ë¢Õß√ŸªÀ≈“¬‡À≈’ˬ¡∑’˧≈⓬°—π ®–‡∑à“°—∫Õ—µ√“ à«π¢Õßæ◊Èπ∑’Ë
¢Õß√Ÿª ’ˇÀ≈’ˬ¡®—µÿ√— ∫π¥â“π§Ÿà¢π“π∑’Ë ¡π—¬°—π
103·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
4. √Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“√ŸªÀπ÷Ëß¡’¥â“𬓫¥â“π≈– 10.8 π‘È« Õ—µ√“ à«π¢Õߧ«“¡¬“«¢Õߥâ“π
¬“«µàÕ¥â“π°«â“߇ªìπ 12 : 5 ®ßÀ“§«“¡¬“«¢Õߥâ“𬓫·≈–¥â“π°«â“ߢÕß√Ÿª ’ˇÀ≈’ˬ¡
º◊πºâ“∑’˧≈⓬°—∫√Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“√Ÿª·√° ·µà¡’æ◊Èπ∑’ˇªìπ 19
¢Õß√Ÿª·√°
«‘∏’∑” „Àâ L1 ·≈– W1 ‡ªì𧫓¡¬“«¢Õߥâ“𬓫·≈–¥â“π°«â“ߢÕß√Ÿª ’ˇÀ≈’ˬ¡
º◊πºâ“√Ÿª·√°
L2 ·≈– W2 ‡ªì𧫓¡¬“«¢Õߥâ“𬓫·≈–¥â“π°«â“ߢÕß√Ÿª ’ˇÀ≈’ˬ¡
º◊πºâ“√Ÿª∑’Ë Õß
√Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“√Ÿª·√°·≈–√Ÿª∑’Ë Õߧ≈⓬°—π ®–‰¥âLW
1
1= ............................... = .....................................
10 8
1
.W
= 125
W1 = ......................................................................
W1 = ..................... Àπ૬
‡π◊ËÕß®“°Õ—µ√“ à«π¢Õßæ◊Èπ∑’Ë¢Õß√ŸªÀ≈“¬‡À≈’ˬ¡∑’˧≈⓬°—π®–‡∑à“°—∫Õ—µ√“ à«π
¢Õßæ◊Èπ∑’Ë¢Õß√Ÿª ’ˇÀ≈’ˬ¡®—µÿ√— ∫π¥â“π§Ÿà∑’Ë ¡π—¬°—π
®–‰¥â«à“æ◊Èπ∑’Ë√Ÿª ’ˇÀ≈’ˬ¡√Ÿª·√°
=L
L
12
22
( )( )
= W
W
12
22
( )( )
¥—ßπ—Èπ 91
=L
L
12
22
( )( )
...................................... = .....................................
...................................... = .....................................
...................................... = .....................................
L2 = ............................... π‘È«
·≈– 91
=W
W
12
22
( )( )
...................................... = .....................................
...................................... = .....................................
...................................... = .....................................
W2 = ............................ π‘È«
æ◊Èπ∑’Ë√Ÿª ’ˇÀ≈’ˬ¡√Ÿª∑’Ë Õß
104 ·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
¥—ßπ—Èπ √Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“√Ÿª∑’Ë Õß¡’¥â“𬓫¬“« ......................π‘È«
·≈–¥â“π°«â“߬“« ......................π‘È«
3. √Ÿª “¡‡À≈’ˬ¡∑’˧≈⓬°—π
∫∑𑬓¡
√Ÿª “¡‡À≈’ˬ¡ Õß√Ÿª§≈⓬°—π °ÁµàÕ‡¡◊ËÕ √Ÿª “¡‡À≈’ˬ¡ Õß√Ÿªπ—Èπ¡’¢π“¥¢Õß¡ÿ¡
‡∑à“°—π‡ªìπ§ŸàÊ “¡§Ÿà
°‘®°√√¡∑’Ë 3
®ßæ‘®“√≥“«à“√Ÿª “¡‡À≈’ˬ¡·µà≈–§ŸàµàÕ‰ªπ’ȧ≈⓬°—πÀ√◊Õ‰¡à ‚¥¬‡µ‘¡™àÕß«à“ß„Àâ ¡∫Ÿ√≥å
«‘∏’∑”
1. P = ............................................. (°”Àπ¥„Àâ)
2. R = ......................................................................
3. Q = ......................................................................
......................................................................
......................................................................
4. ¥—ßπ—Èπ PQRXYZ ......................................................................
......................................................................
QP Z
Y
X
R
105·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
2.
«‘∏’∑”
1. TAS = ................................................... (¡ÿ¡√à«¡)
2. AST = ......................................... ·≈– STA = ...............................................
....................................................................................................................................
....................................................................................................................................
3. ¥—ßπ—Èπ AST ·≈– ABC ‡ªìπ√Ÿª∑’Ë ..................................................................
....................................................................................................................................
3.
«‘∏’∑”
1. QRP = ............................. = 90o ....................................
2. ®“° PQR, PQRRPQ = .............................................................................
3. RPQ = .............................................................................
4. ®“° QSP, QSPPQS = .............................................................................
5. QSP = ........................................ (®“°¢âÕ 4, PQS = 12o)
6. ¥—ßπ—Èπ PQR ·≈– QSP ‡ªìπ√Ÿª∑’Ë ...................................................................
...............................................................................................................................
AS B
T
C
P S
R
Q68o
12o
106 ·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
4.
«‘∏’∑”
1. COA = ...............................................................................................................
2. ACO = ...............................................................................................................
3. OAC = ...............................................................................................................
...............................................................................................................................
4. ¥—ßπ—Èπ ACO ·≈– BDO ‡ªìπ√Ÿª∑’Ë .................................................................
...............................................................................................................................
5.
«‘∏’∑”
1. BCA = ............................................................................................................
2. ABC = ............................................................................................................
3. CAB = ............................................................................................................
............................................................................................................
4. ¥—ßπ—Èπ ABC ·≈– EBD ‡ªìπ√Ÿª∑’Ë ..................................................................
...............................................................................................................................
A
C
OB
D
C
E
D
B
A
107·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
3.1 ¡∫—µ‘¢Õß√Ÿª “¡‡À≈’ˬ¡§≈⓬
∂â“√Ÿª “¡‡À≈’ˬ¡ Õß√Ÿª„¥§≈⓬°—π Õ—µ√“ à«π¢Õߧ«“¡¬“«¢Õߥâ“π§Ÿà∑’Ë
¡π—¬°—π∑ÿ°§Ÿà ‡ªìπÕ—µ√“ à«π∑’ˇ∑à“°—π
°”Àπ¥„Àâ ABC XYZ
a, b ·≈– c ‡ªì𧫓¡¬“«¢Õߥâ“πµ√ߢⓡ¡ÿ¡ A, B ·≈– C µ“¡≈”¥—∫
x, y ·≈– z ‡ªì𧫓¡¬“«¥â“πµ√ߢⓡ¡ÿ¡ X, Y ·≈– Z µ“¡≈”¥—∫
®–‰¥â ax
= by
= cz
®“° ax
= by
®–‰¥â ab
= xy
·≈– by
= cz ®–‰¥â b
c = y
z
°‘®°√√¡∑’Ë 3.1
1. °”Àπ¥√Ÿª “¡‡À≈’ˬ¡·µà≈–§ŸàµàÕ‰ªπ’ȧ≈⓬°—π(1) °”Àπ¥ RPO SQO ®ßÀ“§à“ x ·≈– y
C
A B X Y
Z
ba
c
y x
z
R
P
513
yO 42
x
Q
S
17.5
108 ·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
À“§à“ y, POQO
= ......................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
À“§à“ x, OROS
= ......................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
(2) °”Àπ¥ ABC DEC ®ßÀ“Õ—µ√“ à«π x : y
«‘∏’∑” À“ y ‚¥¬„™â∑ƒ…Æ’∫∑æ’∑“‚°√—
®–‰¥â DC2
= ...............................................
= ...............................................
= ...............................................
= ...............................................
y = ...............................................
®“°√Ÿª ABC DEC
®–‰¥â ABDE
= ...............................................
..................................... = ...............................................
..................................... = ...............................................
x = ...............................................
¥—ßπ—Èπ xy
= ...............................................
= ...............................................
π—Ëπ§◊Õ x : y = ...............................................
D
E
C
178
10
y
A
B
x
109·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
(3) °”Àπ¥ CAT TOP ®ßÀ“§à“¢Õß xy
«‘∏’∑” À“ y ‚¥¬„™â∑ƒ…Æ’∫∑æ’∑“‚°√—
®–‰¥â OP2
= ...............................................
y2
= ...............................................
= ...............................................
= ...............................................
y = ...............................................
®“°√Ÿª CAT TOP
®–‰¥â AT......
= ...............................................
....................................... = ...............................................
....................................... = ...............................................
x = ...............................................
¥—ßπ—Èπ xy = ...............................................
(4) °”Àπ¥ MNO ONT ®ßÀ“§à“ x ·≈– y
P
OT
12 y
6xA
9
C
O
NT 416
M
yx
110 ·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
MNO ONT
®–‰¥â OM......
= ......................... = .....................................
®“° MN
NO= .....................................
................................................................................
NO2
= .....................................
NO = .....................................
®“°√Ÿª “¡‡À≈’ˬ¡¡ÿ¡©‹“° ONT „™â∑ƒ…Æ’∫∑æ’∑“‚°√—
®–‰¥â OT2
= .....................................
................................................................................
................................................................................
x = .....................................
®“° OMOT
= .....................................
...............................................................................
...............................................................................
...............................................................................
y = .....................................
= .....................................
(5) °”Àπ¥ ACE BDE ®ßÀ“§«“¡¬“«¢Õß AC
BC
D
A
2
Ex4
2x2
3x5
111·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
«‘∏’∑” ACE BDE ®–‰¥âAC
BD= ...........................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
x = ...........................................................................
®–‰¥â AC = ...........................................................................
= ...........................................................................
2. °”À𥧫“¡¬“«¢Õߥâ“π¡’Àπ૬‡ªìπ‡´πµ‘‡¡µ√ ®ßÀ“§à“ x ·≈– y ·≈–‡µ‘¡™àÕß«à“ß
„Àâ ¡∫Ÿ√≥å
«‘∏’∑” æ‘®“√≥“ ABC ·≈– DEC
CAB = CDE (µà“߇ªìπ¡ÿ¡©“°)
BCA = ECD (¡ÿ¡√à«¡)
ABC = DEC (√Ÿª “¡‡À≈’ˬ¡ Õß√Ÿª¡’¡ÿ¡‡∑à“°—π 2 §Ÿà ¡ÿ¡§Ÿà∑’ˇÀ≈◊Õ¬àÕ¡‡∑à“°—π)
ABC DEC
D
A B
C
E
y
x
15
20
12
112 ·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
®–‰¥â ACCD
= ABDE
3220
= x15
.................................. = .....................................
x = ................ ‡´πµ‘‡¡µ√
DEC ·≈– ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡¡ÿ¡©“° „™â∑ƒ…Æ’∫∑æ’∑“‚°√— ®–‰¥â
CE2
= .................................
= .....................................
CE = ................. ‡´πµ‘‡¡µ√
·≈– BC2
= ......................................
= ......................................
BC = ................... ‡´πµ‘‡¡µ√
¥—ßπ—Èπ y = ......................................
= ................... ‡´πµ‘‡¡µ√
3. °”À𥧫“¡¬“«¢Õߥâ“π¡’Àπ૬‡ªìπ‡´πµ‘‡¡µ√ ®ßÀ“§à“ x ·≈– y ·≈–‡µ‘¡™àÕß«à“ß
„Àâ∂Ÿ°µâÕß
«‘∏’∑” æ‘®“√≥“ ABC ·≈– DEC
BCA = ECD ...................................................................
CAB = CDE ·≈– CBA = CED .......................................................
....................................................................................................................................
....................................................................................................................................
C
8
2
D
AB
E
14
12
xy
113·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
¥—ßπ—Èπ ABC DEC
®–‰¥â«à“ CACD
= ..........................................
·∑π§à“‡æ◊ËÕÀ“ y, .......................................................................................................
....................................................................................................................................
....................................................................................................................................
·≈– CACD
= ...........................................................................
·∑π§à“‡æ◊ËÕÀ“ x, ......................................................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
4. ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡√ŸªÀπ÷Ëß¡’ ACB = 90o CD AB⊥ ∑’Ë®ÿ¥ D CE ·∫àߧ√÷Ëß AB
∑’Ë®ÿ¥ E ∂â“ CD = 71
5 ‡´πµ‘‡¡µ√ ·≈– AB = 9 ‡´πµ‘‡¡µ√ ®ßÀ“§à“¢Õß x (¥—ß√Ÿª)
«‘∏’∑” ADC ACB ‡æ√“–¡ÿ¡‡∑à“°—π‡ªìπ§ŸàÊ “¡§Ÿà
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
A B
C
D Ex
9
715
114 ·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
À“ AD ®“°√Ÿª “¡‡À≈’ˬ¡¡ÿ¡©“° ADC ¥—ßπ’È
AD2
= ..........................................
.................................. = ..........................................
.................................. = ..........................................
.................................. = ..........................................
¥—ßπ—Èπ AD = ............................... Àπ૬
®“° ADC ACB ®–‰¥â
ACAB
= ..........................................
.................................. = ..........................................
.................................. = ..........................................
.................................. = ..........................................
AB = ............................... Àπ૬
®“°√Ÿª AE = AB2
= ............................... Àπ૬
¥—ßπ—Èπ x = ..........................................
.................................. = ..........................................
.................................. = ..........................................
x = ....................... ‡´πµ‘‡¡µ√
∑ƒ…Æ’∫∑
∂â“Õ—µ√“ à«π¢Õߧ«“¡¬“«¢Õߥâ“π§Ÿà∑’Ë ¡π—¬°—π∑ÿ°§Ÿà¢Õß√Ÿª “¡‡À≈’ˬ¡ Õß√Ÿª‡ªìπ
Õ—µ√“ à«π∑’ˇ∑à“°—π ·≈â«√Ÿª “¡‡À≈’ˬ¡ Õß√Ÿªπ—Èπ‡ªìπ√Ÿª “¡‡À≈’ˬ¡∑’˧≈⓬°—π
115·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
°‘®°√√¡∑’Ë 3.2
®“°√Ÿª “¡‡À≈’ˬ¡ Õß√Ÿª„π·µà≈–¢âÕµàÕ‰ªπ’È ‡ªìπ√Ÿª “¡‡À≈’ˬ¡∑’˧≈⓬°—πÀ√◊Õ‰¡à ‡æ√“–
‡Àµÿ„¥
1.
«‘∏’∑” ‡π◊ËÕß®“° ABRQ
= .........................................................................................
BCQP
= .........................................................................................
CAPR
= .........................................................................................
¥—ßπ—Èπ ABC ·≈– RQP ‡ªìπ√Ÿª∑’Ë ........................... (§≈⓬°—π / ‰¡à§≈⓬°—π)
‡æ√“– ..............................................................................................................
2.
«‘∏’∑” ‡π◊ËÕß®“° TSRP
= .........................................................................................
SQPQ
= .........................................................................................
QTQR
= .........................................................................................
¥—ßπ—Èπ SQT ·≈– PQR ‡ªìπ√Ÿª∑’Ë .......................................................................
‡æ√“– .........................................................................................................................
R Q
P
4.2
34.8
A B
C
3.22
2.8
R
T
3
10
P
8
QS 10
12
2.5
116 ·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
3.
«‘∏’∑” ‡π◊ËÕß®“° YZQR
= .........................................................................................
ZXRP
= .........................................................................................
XYPQ
= .........................................................................................
¥—ßπ—Èπ XYZ ·≈– PQR ‡ªìπ√Ÿª∑’Ë .....................................................................
‡æ√“– .........................................................................................................................
4.
«‘∏’∑” ‡π◊ËÕß®“° PQMP
= .........................................................................................
PRMP
= .........................................................................................
QRPR
= .........................................................................................
¥—ßπ—Èπ PQR ·≈– MPR ‡ªìπ√Ÿª∑’Ë .....................................................................
‡æ√“– .........................................................................................................................
PQ
R
6
4
X12
Y
Z
157.5
3
P
QM
R
1014
124
8
117·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
4. °“√𔉪„™â
°‘®°√√¡∑’Ë 4
1. ABC ·≈– XYZ §≈⓬°—π ¡’æ◊Èπ∑’Ë 32 µ“√“ßπ‘È« ·≈– 60.5 µ“√“ßπ‘È« µ“¡≈”¥—∫ ∂â“
XY = 7.7 π‘È« ®ßÀ“§«“¡¬“«¢Õß AB
«‘∏’∑”
‡π◊ËÕß®“° ABC XYZ ®–‰¥â«à“ Õ—µ√“ à«π¢Õßæ◊Èπ∑’Ë¢Õß√Ÿª “¡‡À≈’ˬ¡∑’Ë
§≈⓬°—π‡∑à“°—∫Õ—µ√“ à«π¢Õß°”≈—ß ÕߢÕߥâ“π∑’Ë ¡π—¬°—π
®–‰¥â æ◊Èπ∑’Ë ABC= AB
XY
2
2
·∑π§à“ .......................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
¥—ßπ—Èπ AB = ........................................... π‘È«
2. √Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“ ABCD ¡’ AB = 15 ‡´πµ‘‡¡µ√ BC = 8 ‡´πµ‘‡¡µ√ O ‡ªìπ
®ÿ¥»Ÿπ¬å°≈“ߢÕ߇ âπ∑·¬ß¡ÿ¡ AC ≈“° OX ⊥ AC ∑’Ë®ÿ¥ O OX æ∫°—∫ AB ∑’Ë®ÿ¥ X
®ßÀ“æ◊Èπ∑’Ë¢Õß√Ÿª “¡‡À≈’ˬ¡ AOX
æ◊Èπ∑’Ë XYZ
X Y
Z
C
B7.7
A
118 ·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
«‘∏’∑”
À“ AO, ABC ¡’ ABC = 90o ®–‰¥â
AC2
= ...........................................................................
AC = ...........................................................................
AO = ........................................................ ‡´πµ‘‡¡µ√
‡π◊ËÕß®“° AOX ABC ®–‰¥â«à“ Õ—µ√“ à«π¢Õßæ◊Èπ∑’Ë¢Õß√Ÿª “¡‡À≈’ˬ¡∑’Ë
§≈⓬°—π ‡∑à“°—∫ Õ—µ√“ à«π¢Õß°”≈—ß ÕߢÕߥâ“π∑’Ë ¡π—¬°—π
®–‰¥â æ◊Èπ∑’Ë¢Õß√Ÿª AOX=
AO2
æ◊Èπ∑’Ë¢Õß√Ÿª ABC AB2
æ◊Èπ∑’Ë¢Õß√Ÿª AOX= ............................................................................
12
158
æ◊Èπ∑’Ë¢Õß√Ÿª AOX= ............................................................................
60
æ◊Èπ∑’Ë¢Õß√Ÿª AOX = ............................................. µ“√“߇´πµ‘‡¡µ√
3. ®“°√Ÿª ABC ¡’æ◊Èπ∑’Ë 24 µ“√“߇´πµ‘‡¡µ√ M ‡ªìπ®ÿ¥∫π AC ∑’Ë∑”„Àâ AM : MC = 7 : 3
≈“° MN AB æ∫ BC ∑’Ë®ÿ¥ N ≈“° AN ®ßÀ“æ◊Èπ∑’Ë¢Õß√Ÿª MAN
B
8
C
O
D
A X15
C
3a
M N
BA
7a
119·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
«‘∏’∑” ABC MNC ‡æ√“–¡’¡ÿ¡‡∑à“°—π‡ªìπ§ŸàÊ “¡§Ÿà ¥—ßπ’È ................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
®–‰¥â«à“ æ◊Èπ∑’Ë¢Õß√Ÿª ABC
= ............................................................................æ◊Èπ∑’Ë¢Õß√Ÿª MNC
....................................................................................................................................
æ◊Èπ∑’Ë¢Õß√Ÿª MNC = ............................................. µ“√“߇´πµ‘‡¡µ√
„Àâ h ‡ªì𧫓¡ Ÿß¢Õß√Ÿª MNC ∑’Ë¡’ MC ‡ªìπ∞“π ®–‰¥â«à“
æ◊Èπ∑’Ë¢Õß√Ÿª MNC = 12
3a h
....................................................................................................................................
....................................................................................................................................
h = .................................................................Àπ૬
æ◊Èπ∑’Ë¢Õß√Ÿª MAN = 12
∞“π Ÿß
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
4. √Ÿª “¡‡À≈’ˬ¡ Õß√Ÿª§≈⓬°—π ¡’æ◊Èπ∑’ˇªìπÕ—µ√“ à«π 13.69 : 16.81 ·≈–√Ÿª„À≠à¡’
§«“¡ Ÿß 10 øÿµ 3 π‘È« √Ÿª “¡‡À≈’ˬ¡√Ÿª‡≈Á°®– Ÿß‡∑à“„¥
«‘∏’∑” „Àâ h1 ·≈– h2 ‡ªì𧫓¡ Ÿß¢Õß√Ÿª “¡‡À≈’ˬ¡√Ÿª„À≠à·≈–√Ÿª‡≈Á° µ“¡≈”¥—∫
h1 = 10 øÿµ 3 π‘È« = ..................................................................................................
æ◊Èπ∑’Ë√Ÿª “¡‡À≈’ˬ¡√Ÿª‡≈Á°= ............................................................................
æ◊Èπ∑’Ë√Ÿª “¡‡À≈’ˬ¡√Ÿª„À≠à....................................................................................................................................
....................................................................................................................................
120 ·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
¥—ßπ—Èπ √Ÿª “¡‡À≈’ˬ¡√Ÿª‡≈Á° Ÿß ............................... øÿµ ............................... π‘È«
5. ‡¥Á°§πÀπ÷Ëß Ÿß 5 øÿµ 4 π‘È« ‡ß“¢Õ߇¢“∑Õ¥¬“« 9 øÿµ 6 π‘È« ∂Ⓡ “∏ßµâπÀπ÷Ë߇°‘¥‡ß“ 38
øÿµ ‡ “∏ß Ÿß‡∑à“‰√
«‘∏’∑”
®“°√Ÿª BC ‡ªì𧫓¡ Ÿß¢Õ߇¥Á° ·≈– AB ‡ªìπ‡ß“¢Õ߇¥Á°
EF ‡ªì𧫓¡ Ÿß¢Õ߇ “∏ß ·≈– DE ‡ªìπ‡ß“¢Õ߇ “∏ß
AB ¬“« 9 øÿµ 6 π‘È« = ...........................................................................
BC ¬“« 5 øÿµ 4 π‘È« = ...........................................................................
DEF ABC
®–‰¥â EFBC
= ...........................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
¥—ßπ—Èπ ‡ “∏ß Ÿß = ..................... øÿµ .................... π‘È«
D E
F
38 øÿµ
5 øÿµ 4 π‘È«
9 øÿµ 6 π‘È«
C
A B
121·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
6. ¥ÿ ‘µ¬◊πÕ¬Ÿà√–À«à“ßÕ“§“√‡√’¬π°—∫‡ “∏ß ‡ªìπ√–¬–∑“ß 114 øÿµ ·≈– 38 øÿµ µ“¡≈”¥—∫
·≈–‡¢“«—¥¡ÿ¡¬°¢÷Èπ∑’ˬեՓ§“√‡√’¬π·≈–‡ “∏߉¥â 42 Õß»“‡∑à“°—π ∂Ⓡ “∏ß Ÿß
35.5 øÿµ ®ßÀ“«à“Õ“§“√‡√’¬π Ÿß‡∑à“‰√
«‘∏’∑”
„Àâ AB ‡ªì𧫓¡ Ÿß¢ÕßÕ“§“√
CD ‡ªì𧫓¡ Ÿß¢Õ߇ “∏ß 35.5 øÿµ
E ‡ªìπ®ÿ¥∑’Ë¥ÿ ‘µ¬◊πÕ¬Ÿà
BE = 114 øÿµ ·≈– DE = 38 øÿµ
ABC CDE ‡æ√“–¡’¡ÿ¡‡∑à“°—π‡ªìπ§ŸàÊ “¡§Ÿà¥—ßπ’È
ABE = CDE = 90o
(°”Àπ¥„Àâ)
BEA = DEC = 42o
(°”Àπ¥„Àâ)
BAE = DCE = ........ (√Ÿª “¡‡À≈’ˬ¡ Õß√Ÿª¡’¡ÿ¡‡∑à“°—π 2 ¡ÿ¡ ¡ÿ¡§Ÿà∑’ˇÀ≈◊Õ
¬àÕ¡‡∑à“°—π)
®–‰¥â ABBE
= ...........................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
¥—ßπ—Èπ Õ“§“√‡√’¬π Ÿß .......................................... øÿµ
42o 42o
35.5
38E D
C
114B
A
122 ·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
7.
®“°√Ÿª AB ‡ªì𧫓¡ Ÿß¢Õß«à“«∑’Ë≈Õ¬Õ¬Ÿà‡Àπ◊Õæ◊Èπ¥‘π
CD ‡ªì𧫓¡ Ÿß¢Õßµâπ‰¡â 10 øÿµ
AE ‡ªì𧫓¡¬“«¢Õß “¬ªÉ“π 650 øÿµ
ED ‡ªìπ√–¬–Àà“ß®“° ¡æ≈∂÷ß‚§πµâπ‰¡â 24 øÿµ
®ßÀ“§«“¡ Ÿß¢Õß«à“«∑’Ë≈Õ¬Õ¬Ÿà‡Àπ◊Õæ◊Èπ¥‘π
«‘∏’∑” À“§«“¡¬“«¢Õß CE
®“° CED ‡ªìπ√Ÿª “¡‡À≈’ˬ¡¡ÿ¡©“°
®–‰¥â«à“ CE2
= ...........................................................................
= ...........................................................................
= ...........................................................................
= ...........................................................................
CE = ...........................................................................
®“°√Ÿª CED AEB ‡æ√“–«à“¡’¡ÿ¡‡∑à“°—π‡ªìπ§ŸàÊ “¡§Ÿà ¥—ßπ’È
EDC = ......................................................................................................
CED = ......................................................................................................
DCE = ......................................................................................................
¥—ßπ—Èπ CDAB
= ..........................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
¥—ßπ—Èπ «à“«Õ¬Ÿà Ÿß®“°æ◊Èπ¥‘π ....................................øÿµ
650'
10'
24'E
B
A
C
D
123·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
8. ∫—π‰¥¬“« 4 ‡¡µ√ ª≈“¬∫—π‰¥æ“¥Õ¬Ÿà∑’Ë¢Õ∫√—È«∫â“πæÕ¥’ ‡¡◊ËÕ™à“ß∑“ ’¢÷Èπ‰ª‰¥â 23
¢Õß
§«“¡¬“«¢Õß∫—π‰¥ ‡¢“∑”·ª√ß∑“ ’µ° ∂â“®ÿ¥∑’Ë·ª√ß∑“ ’µ°≈ß¡“∑’Ëæ◊Èπ¥‘πÀà“ß®“°
√—È«∫â“π 25
‡¡µ√ Õ¬“°∑√“∫«à“ ª≈“¬∫—π‰¥∑’Ë·µ–°—∫æ◊Èπ¥‘πÀà“ß®“°√—È«∫â“π‡∑à“‰√
«‘∏’∑” „Àâ AB ‡ªìπ√—È«∫â“π AC ‡ªìπ∫—π‰¥¬“« 4 ‡¡µ√ ®ÿ¥ D ·∫àß∫—π‰¥ÕÕ°‡ªìπ 23
¢Õߧ«“¡¬“«¢Õß∫—π‰¥ ®ÿ¥ E ‡ªìπ®ÿ¥∑’Ë·ª√ß∑“ ’µ°≈ß¡“ BE = 25
‡¡µ√
„Àâ CE = a ‡¡µ√
∑’Ë∫—π‰¥ CD = 23
4
= ................... ‡¡µ√
DEC ABC ®–‰¥âCEBC
= ...........................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
¥—ßπ—Èπ ª≈“¬∫—π‰¥∑’Ë·µ–°—∫æ◊ÈπÀà“ß®“°√—È« .................................................. ‡¡µ√
4 ¡.
EB
A
D
a C25
¡.
124 ·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
9. ®“°√Ÿª ®ßÀ“§«“¡°«â“ߢÕß·¡àπÈ”
«‘∏’∑”
®“°√Ÿª AB ‡ªì𧫓¡°«â“ߢÕß·¡àπÈ”
À“ BC, CBE ‡ªìπ√Ÿª “¡‡À≈’ˬ¡¡ÿ¡©“°
®–‰¥â BC2
= ..............................................
= ..............................................
= ..............................................
= ..............................................
BC = ..............................................
æ‘®“√≥“ ABD ·≈– CBE §≈⓬°—π ‡æ√“–¡’¡ÿ¡‡∑à“°—π‡ªìπ§ŸàÊ “¡§Ÿà
¥—ßπ’È BAD = ....................................................................................
ADB = ....................................................................................
ABD = ....................................................................................
®–‰¥â ABBC
= ....................................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
¥—ßπ—Èπ ·¡àπÈ”°«â“ß ................................................ øÿµ
10. ®“°‡™‘ßÀÕ§Õ¬ ¡ÿ¡¬°¢÷Èπ¢Õߪ≈“¬‡ “À‘πµâπÀπ÷Ë߇ªìπ
65o ·≈–®“°¬Õ¥ÀÕ§Õ¬ ÷Ëß Ÿß 48 øÿµ ¡ÿ¡¬°¢÷Èπ¢Õß
ª≈“¬‡ “À‘πµâππ—Èπ‡ªìπ 25o ®ßÀ“§«“¡ Ÿß¢Õ߇ “À‘π
∂Ⓡ™‘ßÀÕ§Õ¬·≈–‡ “À‘πÀà“ß°—π 18 øÿµ
«‘∏’∑” „Àâ AB ‡ªì𧫓¡ Ÿß¢Õ߇ “À‘π
CD ‡ªì𧫓¡ Ÿß¢ÕßÀÕ§Õ¬ Ÿß 48 øÿµ
CE ‡ªìπ‡ âπ√–¥—∫®“° C ∂÷ß AB
4 øÿµ
12 øÿµ
100 øÿµ
CE
B
A D
B
E
A
C
D18 øÿµ
48 øÿµ
65o
25o
x
48 øÿµ
125·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
®–‰¥â CD = EB = 48 øÿµ
·≈– BD = CE = 18 øÿµ
°”Àπ¥„Àâ AB ‡∑à“°—∫ 48 x øÿµ
ACE DAB ‡æ√“–¡’¡ÿ¡‡∑à“°—π‡ªìπ§ŸàÊ “¡§Ÿà ¥—ßπ’È
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®–‰¥â«à“ AEBD
= ...........................................................................
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¥—ßπ—Èπ ‡ “À‘π Ÿß .............................................. øÿµ
126 ·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
1. ®ßæ‘®“√≥“¢âÕ§«“¡µàÕ‰ªπ’È
(1) √Ÿª “¡‡À≈’ˬ¡ Õß√Ÿª∑’ˇ∑à“°—π∑ÿ°
ª√–°“√ ®–‡ªìπ√Ÿª “¡‡À≈’ˬ¡∑’˧≈⓬°—π
(2) √Ÿª “¡‡À≈’ˬ¡ Õß√Ÿª∑’˧≈⓬°—π ®–
‡ªìπ√Ÿª “¡‡À≈’ˬ¡∑’ˇ∑à“°—π∑ÿ°ª√–°“√
(3) √Ÿª “¡‡À≈’ˬ¡ Õß√Ÿª∑’Ë¡’¢π“¥¢Õß
¡ÿ¡‡∑à“°—π Õß¡ÿ¡ √Ÿª “¡‡À≈’ˬ¡ Õß
√Ÿªπ’ȧ≈⓬°—π
¢âÕ§«“¡„¥∂Ÿ°µâÕß
1. ¢âÕ (1) ·≈– ¢âÕ (2)
2. ¢âÕ (1) ·≈– ¢âÕ (3)
3. ¢âÕ (2) ·≈– ¢âÕ (3)
4. ∂Ÿ°µâÕß∑ÿ°¢âÕ
2. ®ßæ‘®“√≥“¢âÕ§«“¡µàÕ‰ªπ’È
(1) √ŸªÀ°‡À≈’ˬ¡ Õß√Ÿª ¡’¡ÿ¡‡∑à“°—π
¡ÿ¡µàÕ¡ÿ¡ √ŸªÀ°‡À≈’ˬ¡ Õß√Ÿªπ’ȧ≈⓬°—π
(2) √ŸªÀ°‡À≈’ˬ¡ Õß√Ÿª ¡’Õ—µ√“ à«π
¢Õߥâ“π‡∑à“°—π∑ÿ°§Ÿà √ŸªÀ°‡À≈’ˬ¡ Õß
√Ÿªπ’ȧ≈⓬°—π
¢âÕ§«“¡„¥∂Ÿ°µâÕß
1. ¢âÕ (1)
2. ¢âÕ (2)
3. ¢âÕ (1) ·≈– ¢âÕ (2)
4. º‘¥∑—ÈߢâÕ (1) ·≈– ¢âÕ (2)
3.
®“°√Ÿª xy ‡∑à“°—∫‡∑à“„¥
1. 154.45 Àπ૬ 2. 10.55 Àπ૬
3. 8.55 Àπ૬ 4. 6.45 Àπ૬
4. AD = 3 ‡´πµ‘‡¡µ√ ·≈– BD = 12
‡´πµ‘‡¡µ√ CD ¬“«‡∑à“„¥
1. 35
‡´πµ‘‡¡µ√
2. 36
‡´πµ‘‡¡µ√
3. 5 ‡´πµ‘‡¡µ√
4. 6 ‡´πµ‘‡¡µ√
5. ∂â“ ABC FDE ·≈â« m + n ‡∑à“°—∫
‡∑à“„¥
1. 7 2 Àπ૬ 2. 10 2 Àπ૬
3. 12 2 Àπ૬ 4. 13 2 Àπ૬
·∫∫∑¥ Õ∫º≈ —¡ƒ∑∏‘Ï∑“ß°“√‡√’¬πª√–®”Àπ૬
A B
C
18
4m
n
D E
F
12 3m
10
C
A D3 12B
x y
4.5
311.5
„Àâπ—°‡√’¬π‡≈◊Õ°§”µÕ∫∑’Ë∂Ÿ°µâÕß
127·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
6. ®“°√Ÿª RS ·∫àߧ√÷Ëß PRQ ∂â“ PQ = 9
‡´πµ‘‡¡µ√ QR = 7 ‡´πµ‘‡¡µ√ RP = 5
‡´πµ‘‡¡µ√ ·≈â« SQ ¬“«‡∑à“‰√
1. 5 14
‡´πµ‘‡¡µ√
2. 5 34
‡´πµ‘‡¡µ√
3. 6 12 ‡´πµ‘‡¡µ√
4. ¢âÕ¡Ÿ≈‰¡à‡æ’¬ßæÕ„π°“√À“§”µÕ∫
7. PQR ¡’¥â“π PQ = 1.8 ‡´πµ‘‡¡µ√QR = 8 ‡´πµ‘‡¡µ√ PR = 8.2 ‡´πµ‘‡¡µ√
∂â “ PQR §≈⓬°—∫ P´Q´R ·≈–
‡ âπ√Õ∫√Ÿª¢Õß P´Q´R´ ¬“« 31.5
‡´πµ‘‡¡µ√ ¥â“π∑’ˬ“«∑’Ë ÿ¥¢Õß P´Q´R´
‡∑à“°—∫‡∑à“„¥
1. 14 ‡´πµ‘‡¡µ√2. 14.25 ‡´πµ‘‡¡µ√
3. 14.35 ‡´πµ‘‡¡µ√
4. 14.45 ‡´πµ‘‡¡µ√8. ®“°√Ÿª ‡ âπ√Õ∫√Ÿª
¢Õß DBE ‡∑à“°—∫
4 2 5 ‡´πµ‘‡¡µ√DE ¬“«‡∑à“„¥
1. 4 2 ‡´πµ‘‡¡µ√
2. 5 ‡´πµ‘‡¡µ√3. 5 2 ‡´πµ‘‡¡µ√
4. 6 ‡´πµ‘‡¡µ√
9. ®“°√Ÿª ABC PBQ §à“¢Õß xy
‡∑à“°—∫‡∑à“‰√
1. 15 Àπ૬ 2. 17 Àπ૬
3. 19 Àπ૬ 4. 21 Àπ૬
10. √Ÿª “¡‡À≈’ˬ¡ ABC ¡’æ◊Èπ∑’Ë 25.6
µ“√“߇´πµ‘‡¡µ√ ≈“° XY ¢π“π°—∫
BC µ—¥ AB ·≈– AC ∑’Ë®ÿ¥ X ·≈– Y
µ“¡≈”¥—∫ ∑”„Àâ AX : XB = 5 : 3
®ßÀ“æ◊Èπ∑’Ë¢Õß√Ÿª “¡‡À≈’ˬ¡ AXY
1. 10 µ“√“߇´πµ‘‡¡µ√
2. 10.25 µ“√“߇´πµ‘‡¡µ√
3. 12 µ“√“߇´πµ‘‡¡µ√
4. 12.25 µ“√“߇´πµ‘‡¡µ√
11. √Ÿª “¡‡À≈’ˬ¡ ABC ¡’ ABC = 90o
AB = 18 ‡´πµ‘‡¡µ√ BC = 9 ‡´πµ‘‡¡µ√
X ‡ªìπ®ÿ¥°÷Ëß°≈“ߢÕß BC ∂â“ XY
µ—Èß©“°°—∫ AC ∑’Ë®ÿ¥ Y ·≈â« XY ¬“«
°’ˇ´πµ‘‡¡µ√
1. 95
‡´πµ‘‡¡µ√
2. 9 55
‡´πµ‘‡¡µ√
3. 910
‡´πµ‘‡¡µ√
4. 9 510
‡´πµ‘‡¡µ√
R
PS
Q
1 2
C
14
A BE
D
A
C
BQ
Py
x
2
43
2
10 2
128 ·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
12. ®“°√Ÿª ABCD ‡ªìπ√Ÿª ’ˇÀ≈’ˬ¡§“ßÀ¡Ÿ
EF // AB ·≈– AE : ED = 2 : 5 ∂â“
EF = 36 ‡´πµ‘‡¡µ√ ·≈– AB = 30
‡´πµ‘‡¡µ√ ·≈â« CD ¬“«‡∑à“„¥
1. 66 ‡´πµ‘‡¡µ√ 2. 60 ‡´πµ‘‡¡µ√
3. 56 ‡´πµ‘‡¡µ√ 4. 51 ‡´πµ‘‡¡µ√
13. ®“°√Ÿª m : n ‡∑à“°—∫‡∑à“„¥
1. 9 : 7 2. 12 : 7
3. 13 : 8 4. 15 : 8
14. ¥«ß‰ø¥«ßÀπ÷ËßÕ¬ŸàÀà“ß®“°ºπ—ßÀâÕß 10
øÿµ √Ÿª¿“æ ’Ë‡À≈’ˬ¡º◊πºâ“√ŸªÀπ÷Ëß¡’
¥â“𬓫 16 π‘È« ·≈–¥â“π°«â“ß 12 π‘È«
«“ßÀà“ß®“°ºπ—ßÀâÕß 6 øÿµ ∂Ⓡߓ¢Õß
√Ÿª¿“æ∑’˺π—ßÀâÕ߇ªìπ√Ÿª ’ˇÀ≈’ˬ¡º◊π
ºâ“®–¡’æ◊Èπ∑’ˇ∑à“„¥
1. 6 13 µ“√“ßøÿµ 2. 6 2
3 µ“√“ßøÿµ
3. 8 14
µ“√“ßøÿµ 4. 8 12
µ“√“ßøÿµ
15. ·¡Á§¬◊πÕ¬Ÿà∫πæ◊Èπ√“∫ ¡Õ߇ÀÁπ¬Õ¥
µâπ π ·≈–¬Õ¥‡ “∏ßÕ¬Ÿà„π·π«‡ âπµ√ß
‡¥’¬«°—π ¥—ß√Ÿª
∂â“·¡Á§¬◊πÀà“ß®“°µâπ π 20 ‡¡µ√ ·≈–
µâπ πÕ¬ŸàÀà“ß®“°‡ “∏ß 12 ‡¡µ√ ∂â“
‡ “∏ß Ÿß 15 ‡¡µ√ µâπ π Ÿß°’ˇ¡µ√
1. 9 14
‡¡µ√ 2. 9 12
‡¡µ√
3. 9 18
‡¡µ√ 4. 9 38
‡¡µ√
16. ®“°√Ÿª ·¡àπÈ”¡’§«“¡°«â“߇∑à“„¥ ¡’
Àπ૬§«“¡¬“«‡ªìπøÿµ
1. 45 2 øÿµ 2. 45 3 øÿµ
3. 50 2 øÿµ 4. 50 3 øÿµ
17. ª“¬‡ªî¥√â “π¢“¬
‡§√◊ËÕߪ√–¥—∫ ®÷ß
ÕÕ°·∫∫Õÿª°√≥å
·¢«π‡§√◊ËÕߪ√–¥—∫
‡ªìπ‰¡â°≈÷ß°≈¡ 5
Õ—π ∑’˵հµ‘¥°—∫
·°π‰¡â„Àâ¢π“π°—π
D C
A B
E F36
30
·¡àπÈ”
C
BA
1504
DE
8
B C18
E
F
H
A
Dm
n5
4
·¡Á§
129·∫∫Ωñ°§≥‘µ»“ µ√åæ◊Èπ∞“π ¡.3 ¿“§‡√’¬π∑’Ë 1
·≈–Àà“߇∑à“°—π¥—ß√Ÿª ∂Ⓣ¡â°≈÷ß°≈¡
Õ—π∑’ˬ“«∑’Ë ÿ¥¬“« 12 π‘È« ª“¬®–µâÕß
´◊ÈÕ‰¡â°≈÷ß°≈¡¬“«°’Ëøÿµ ∂⓵âÕß∑”
Õÿª°√≥å·¢«π·∫∫π’È 3 ™‘Èπ
1. 3 øÿµ 2. 6 øÿµ
3. 9 øÿµ 4. 12 øÿµ
18. ∂“π∑’Ë Õß·ÀàßÀà“ß°—π 200 øÿµ ·≈–
Õ¬Ÿà∑“ß∑‘»µ–«—πÕÕ°¢Õߪ√“ “∑·Ààß
Àπ÷Ëß ¡’¡ÿ¡¬°¢÷Èπ¢Õ߬եª√“ “∑‡ªìπ
30o ·≈– 60
o ª√“ “∑ Ÿß°’Ëøÿµ
1. 100 øÿµ 2. 141.40 øÿµ
3. 173.20 øÿµ 4. 185.50 øÿµ
19. ®“°¬Õ¥ ÿ¥¢ÕßÀπ⓺“ ÷Ëß Ÿß 150 øÿµ
‡¥Á°§πÀπ÷Ëß¡Õ߇ÀÁπ‡√◊Õ A ·≈– B ‡ªìπ
¡ÿ¡°¥≈ß 15o ·≈– 75
o °—∫·π«√–¥—∫
µ“¡≈”¥—∫ ∂Ⓡ√◊Õ B Õ¬ŸàÀà“ß®“°
Àπ⓺“ª√–¡“≥ 40 øÿµ ‡√◊Õ A Õ¬ŸàÀà“ß
®“°‡√◊Õ B °’Ëøÿµ
1. 652.50 øÿµ 2. 532.50 øÿµ
3. 530.50 øÿµ 4. 522.50 øÿµ
20. ¿Ÿ‡¢“≈Ÿ°Àπ÷Ëß Ÿß®“°æ◊Èπ¥‘π 4,900 øÿµ
®“°®ÿ¥ A ∫πæ◊Èπ¥‘π ¡ÿ¡¬°¢÷Èπ¢Õß
¬Õ¥‡¢“‡ªìπ 60o 18´ ¡’∫Õ≈≈Ÿπ≈Ÿ°Àπ÷Ëß
≈Õ¬¢÷Èπ®“° A µ√ߢ÷Èπ‰ª„π·π«¥‘Ëߥ⫬
Õ—µ√“‡√Á« ¡Ë”‡ ¡Õ ‡¡◊ËÕ¢÷Èπ‰ª‰¥â 5 π“∑’
ºŸâÕ¬Ÿà„π∫Õ≈≈Ÿπ —߇°µ‡ÀÁπ¡ÿ¡¬°¢÷Èπ¢Õß
¬Õ¥‡¢“‡ªìπ 29o 42´ Õ—µ√“‡√Á«¢Õß
∫Õ≈≈Ÿπ‡ªìπ‡∑à“„¥ ∂â“®ÿ¥ A Õ¬ŸàÀà“ß
®“°‡™‘߇¢“ 2,800 øÿµ
1. 8 ‰¡≈åµàÕ™—Ë«‚¡ß
2. 7.5 ‰¡≈åµàÕ™—Ë«‚¡ß
3. 7 ‰¡≈åµàÕ™—Ë«‚¡ß
4. 6.5 ‰¡≈åµàÕ™—Ë«‚¡ß
150 øÿµ
40 øÿµ B15o
15o
75o
75o
A
30o60o
200 øÿµ
2,800 øÿµ A
60 o18'
29o42'
