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1. √Ÿª‡√¢“§≥‘µ∑’˧≈⓬°—𠧫“¡§≈⓬ 4 √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)

097-114 M3/4 P1 · 2016-02-02 · 102 ·∫∫Ωñ°§≥ ‘µ»“ µ√ åæ Èπ∞“π ¡.3 ¿“§‡√ ’¬π∑ ’Ë 1 «‘∏’∑” ‡π ËÕß®“° abcd wxyz

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Text of 097-114 M3/4 P1 · 2016-02-02 · 102 ·∫∫Ωñ°§≥ ‘µ»“ µ√ åæ Èπ∞“π ¡.3...

Page 1: 097-114 M3/4 P1 · 2016-02-02 · 102 ·∫∫Ωñ°§≥ ‘µ»“ µ√ åæ Èπ∞“π ¡.3 ¿“§‡√ ’¬π∑ ’Ë 1 «‘∏’∑” ‡π ËÕß®“° abcd wxyz

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)

Page 2: 097-114 M3/4 P1 · 2016-02-02 · 102 ·∫∫Ωñ°§≥ ‘µ»“ µ√ åæ Èπ∞“π ¡.3 ¿“§‡√ ’¬π∑ ’Ë 1 «‘∏’∑” ‡π ËÕß®“° abcd wxyz

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)

Page 3: 097-114 M3/4 P1 · 2016-02-02 · 102 ·∫∫Ωñ°§≥ ‘µ»“ µ√ åæ Èπ∞“π ¡.3 ¿“§‡√ ’¬π∑ ’Ë 1 «‘∏’∑” ‡π ËÕß®“° abcd wxyz

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

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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

Page 5: 097-114 M3/4 P1 · 2016-02-02 · 102 ·∫∫Ωñ°§≥ ‘µ»“ µ√ åæ Èπ∞“π ¡.3 ¿“§‡√ ’¬π∑ ’Ë 1 «‘∏’∑” ‡π ËÕß®“° abcd wxyz

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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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....................................................................................................................................

....................................................................................................................................

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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

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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

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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

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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

¡.

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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

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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

....................................................................................................................................

....................................................................................................................................

....................................................................................................................................

®–‰¥â«à“ AEBD

= ...........................................................................

....................................................................................................................................

....................................................................................................................................

....................................................................................................................................

....................................................................................................................................

....................................................................................................................................

....................................................................................................................................

....................................................................................................................................

....................................................................................................................................

....................................................................................................................................

....................................................................................................................................

¥—ßπ—Èπ ‡ “À‘π Ÿß .............................................. øÿµ

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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

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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

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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§

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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®“°

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