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·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1 75
°“√·ª≈ß∑“߇√¢“§≥‘µ
4
1. °“√‡≈◊ËÕπ¢π“π
°‘®°√√¡∑’Ë 1
1. ®ßÕ∏‘∫“¬√Ÿª∑’ˉ¥â®“°°“√‡≈◊ËÕπ¢π“π√Ÿªµâπ·∫∫∑’Ë°”Àπ¥µàÕ‰ªπ’È
(1)
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·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 176
(2)
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(3)
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(4)
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·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1 77
(5)
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(6)
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(7)
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·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 178
2. ®ßÕ∏‘∫“¬°“√‡≈◊ËÕπ¢π“π¢Õß√Ÿª‡≈◊ËÕπ¢π“π®“° —≠≈—°…≥åµàÕ‰ªπ’È
(1) 4
2
....................................................................................................................
(2)
4
2
....................................................................................................................
(3)
4
2
....................................................................................................................
(4)
4
2
....................................................................................................................
3. ®ßÀ“ —≠≈—°…≥宓°°“√‡≈◊ËÕπ¢π“π‡¡◊ËÕ°”Àπ¥®ÿ¥¢Õß°“√‡≈◊ËÕπ¢π“π¥—ßπ’È
(1) ®“° (1, 3) ‰ª¬—ß (3, 1) ................................................................................................
(2) ®“° (2, 3) ‰ª¬—ß (3, 2) ..........................................................................................
(3) ®“° (1, 3) ‰ª¬—ß (3, 1) ..........................................................................................
(4) ®“° (2, 3) ‰ª¬—ß (3, 2) ..........................................................................................
(5) ®“° (1, 3) ‰ª¬—ß (3, 1) .......................................................................................
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1 79
C(7, 2)
A(2, 1)
B(4, 3)
2
O
4
2468
2
4
2 4 6 8X
Y
4. °”Àπ¥√Ÿª ’ˇÀ≈’ˬ¡ ABCD „Àâπ—°‡√’¬π‡≈◊ËÕπ¢π“π√Ÿª ’ˇÀ≈’ˬ¡ ABCD „π√–π“∫ XY ¥—ßπ’È
‡≈◊ËÕπ®ÿ¥ A(2, 1), B(5, 2), C(1, 3) ·≈– D(4, 4) ‰ª¬—ß®ÿ¥ A(0, 4), B(3, 3), C(1, 2)
·≈– D(2, 1) µ“¡≈”¥—∫ ·≈–®ßµÕ∫§”∂“¡µàÕ‰ªπ’È
√Ÿªµâπ·∫∫§◊Õ ........................... ·≈–√Ÿª∑’ˉ¥â®“°°“√‡≈◊ËÕπ¢π“π§◊Õ ..................................
√Ÿª ’ˇÀ≈’ˬ¡ Õß√Ÿª‡∑à“°—π∑ÿ°ª√–°“√À√◊Õ‰¡à .......................................................................
5. ®ß‡¢’¬π¿“殓°°“√‡≈◊ËÕπ¢π“π√Ÿª∑’Ë°”Àπ¥„Àâ„π·µà≈–¢âÕµàÕ‰ªπ’È
(1) ‡≈◊ËÕπ¢π“π°—∫·°π X ‰ª∑“ß â“¬¡◊Õ 10 Àπ૬
Y
X
C(1, 3)
D(4, 4)
B(5, 2)
A(2, 1)
A(0, 4)
B(3, 3)C(1, 2)
D(2, 1)O
2
4
24 2 4 6
2
4
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 180
(2) ‡≈◊ËÕπ¢π“π°—∫·°π Y ≈߉ª 3 Àπ૬
(3) ‡≈◊ËÕπ¢π“π°—∫·°π X ‰ª∑“ߴ⓬¡◊Õ 5 Àπ૬ ·≈–‡≈◊ËÕπ≈ß°—∫·°π Y 3 Àπ૬
6. ®ßÀ“®ÿ¥∑’ˉ¥â®“°°“√‡≈◊ËÕπ¢π“π¢Õß®ÿ¥ (2, 1) ≈ß∫π√–π“∫ X, Y ‡¥’¬«°—π
(1) ¿“¬„µâ°“√‡≈◊ËÕπ¢π“π 3
2
(2) ¿“¬„µâ°“√‡≈◊ËÕπ¢π“π
3
2
(3) ¿“¬„µâ°“√‡≈◊ËÕπ¢π“π
3
2
(4) ¿“¬„µâ°“√‡≈◊ËÕπ¢π“π
3
2
Y
B(3, 2)
A(3, 1)
C(1, 1)
O24
2
2 4 6
2
4
2 O
A(1, 3)
B(4, 2)
C(1, 1)
Y
46
2
4
2
2 4 6
X
X
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1 81
7. ®ß‡¢’¬π¿“殓°°“√‡≈◊ËÕπ¢π“π√Ÿª∑’Ë°”Àπ¥„Àâ„π·µà≈–¢âÕµàÕ‰ªπ’È
(1) 3
1
(2)
3
1
X
Y
(2, 1)
1 O23456 1 2 3 4 5 6
1
2
3
4
1
2
3
Y
X
C(2, 2)
B(1, 0)
A(4, 2)
O1 1
1
2
2 3 4 5 623456
2
3
D(3, 4) C(4, 4)
B(5, 2)
A(2, 1)
O1 1
1
123456
2
2 3 4 5 6
2
3
4
5
Y
X
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 182
(3)
3
1
(4)
3
1
D(1, 6)
C(4, 4)
D(5, 2)
E(3, 3)
O1 1
1
23456 2 3 4 5 6
2
3
4
5
6
7
1
2
3
4
Y
X
Y
X
A(1, 3)
B(2, 2)
C(3, 0)
D(4, 1)
O1 11
1
2
3
2
3
4
2 3 4 5 623456
A
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1 83
8. °”Àπ¥√Ÿª “¡‡À≈’ˬ¡ ABC ¡’®ÿ¥¬Õ¥ A(3, 3), B(1, 2) ·≈– C(5, 0) µ“¡≈”¥—∫
®ß‡¢’¬π¿“æ∑’ˇ°‘¥®“°°“√‡≈◊ËÕπ¢π“π√Ÿª “¡‡À≈’ˬ¡ ABC ¿“¬„µâ°“√‡≈◊ËÕπ¢π“π
3
2
9. °”Àπ¥√Ÿª ’ˇÀ≈’ˬ¡ ABCD ∑’Ë¡’®ÿ¥¬Õ¥ A(4, 2), B(2, 2), C(1, 2) ·≈– D(3, 0) µ“¡
≈”¥—∫ ®ß‡¢’¬π√Ÿª∑’ˉ¥â®“°°“√‡≈◊ËÕπ¢π“π√Ÿª ’ˇÀ≈’ˬ¡ ABCD ¿“¬„µâ°“√‡≈◊ËÕπ¢π“π
4
2
O1 1
1
2
Y
X
2
3
23456 2 3 4 5 61
3
4
A(3, 3)
C(5, 0)
B(1, 2)
O1 1
1
2
X
2
3
2345 2 3 4 5 61
3
4
Y
7
B(2, 2) C(1, 2)
D(3, 0)
A(4, 2)
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 184
°‘®°√√¡∑’Ë 2.1
1. „Àâπ—°‡√’¬π≈“°·°π ¡¡“µ√¢Õß√ŸªµàÕ‰ªπ’È·≈–∫Õ°¥â«¬«à“·µà≈–√Ÿª¡’·°π ¡¡“µ√°’Ë·°π
(1) (2) (3)
.............................. .............................. ..............................
(4) (5) (6)
.............................. .............................. ..............................
2. °“√ –∑âÕπ
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1 85
(7) (8) (9)
.............................. .............................. ..............................
(10)
......................................
2. ®ß«“¥¿“æ –∑âÕπ¢Õß√ŸªµàÕ‰ªπ’È
(1) (2)
(3) (4)
‡ âπ –∑âÕπ
‡ âπ –∑âÕπ
‡ âπ –∑âÕπ
‡ âπ –∑âÕπ
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 186
(5) (6)
(7) (8)
(9) (10)
3. ®ß≈Õ°√ŸªµàÕ‰ªπ’È·≈⫇¢’¬π‡ âπ –∑âÕπ„π·µà≈–√Ÿª
(1) (2)
(3) (4)
‡ âπ –∑âÕπ
‡ âπ –∑âÕπ
‡ âπ –∑âÕπ
‡ âπ –∑âÕπ
‡ âπ –∑âÕπ
‡ âπ –∑âÕπ
‡ âπ –∑âÕπ
‡ âπ –∑âÕπ
‡ âπ –∑âÕπ
‡ âπ –∑âÕπ
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1 87
°‘®°√√¡∑’Ë 2.2
1. √Ÿª«ß°≈¡∑’Ë¡’®ÿ¥»Ÿπ¬å°≈“ßÕ¬Ÿà∑’Ë®ÿ¥ (2, 2) ·≈–¡’√—»¡’ 2 Àπ૬ –∑âÕπ¥â«¬‡ âπµ√ß∑’Ë·∫àß
§√÷Ëߧ«Õ¥√—πµå∑’Ë 1 ®ß«“¥¿“æ·≈–À“æ‘°—¥¢Õß®ÿ¥»Ÿπ¬å°≈“ß
2. °”Àπ¥„Àâ√Ÿª ’ˇÀ≈’ˬ¡ ABCD ¡’ ‡ªìπ‡ âπ –∑âÕπ ®ß‡¢’¬π√Ÿª –∑âÕπ¢Õß√Ÿª ’ˇÀ≈’ˬ¡ ABCD
æ√âÕ¡∑—Èß∫√√¬“¬¢—ÈπµÕπ«‘∏’ √â“ߥ⫬
Y
X
4
2
4 2
6
8
42 6O
l D
C
X
A
B
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 188
............................................................................................................................................
............................................................................................................................................
............................................................................................................................................
............................................................................................................................................
............................................................................................................................................
............................................................................................................................................
............................................................................................................................................
............................................................................................................................................
3. √Ÿª ’ˇÀ≈’ˬ¡ ABCD –∑âÕπ¥â«¬‡ âπµ√ß∑’Ë¢π“π°—∫·°π Y ·≈–µ—¥·°π X ∑’Ë®ÿ¥ (2, 0)
®ß«“¥¿“æ·≈–À“æ‘°—¥¢Õß®ÿ¥ –∑âÕπ¢Õß®ÿ¥ A(2, 4), B(4, 4), C(6, 0) ·≈–
(12, 0)
4. √Ÿª ’ˇÀ≈’ˬ¡¡ÿ¡©“° ABCD –∑âÕπ¥â«¬‡ âπ∑’Ë¢π“π°—∫·°π X ·≈–µ—¥·°π Y ∑’Ë®ÿ¥ (2, 0)
®ß«“¥¿“æ ·≈–À“æ‘°—¥¢Õß®ÿ¥ –∑âÕπ¢Õß®ÿ¥ A(2, 1), B(5, 1), C(5, 6) ·≈– D(2, y)
‡ âπ –∑âÕπ
Y
X
A
B
CD
(4, 4)
(2, 4)
4
2
12 10 8 6 4 2 O 2 6 8 10 12 14 16 18
6
2
4
‡ âπ –∑âÕπ
A B
CD
2O2
Y
X
4
6
8
10
246810 4 6 8 10 12 14 16
2
4
6
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1 89
5. √Ÿª ’ˇÀ≈’ˬ¡¡ÿ¡©“° ABCD –∑âÕπ¥â«¬‡ âπµ√ß∑’Ë¢π“π°—∫·°π Y ·≈–µ—¥·°π X ∑’Ë®ÿ¥ (2, 0)
®ß«“¥¿“æ·≈–À“æ‘°—¥¢Õß®ÿ¥ –∑âÕπ¢Õß®ÿ¥ A(3, 2), B(7, 2), C(9, 3) ·≈– D(5, 3)
6. √Ÿª ’ˇÀ≈’ˬ¡ ABCD –∑âÕπ¥â«¬‡ âπµ√ß∑’Ë¢π“π°—∫·°π X ·≈–µ—¥·°π Y ∑’Ë®ÿ¥ (0, 1)
®ß«“¥¿“æ·≈–À“æ‘°—¥¢Õß®ÿ¥ –∑âÕπ¢Õß®ÿ¥ A(3, 1), B(6, 2), C(4, 8) ·≈– D(0, 6)
Y
X
C(4, 8)
D(0, 6)
B(6, 2)A(3, 1)
2
2
4
6
8
4
6
8
10
6810 O24 2 4 6 8 10 12 14 16
‡ âπ –∑âÕπ
‡ âπ –∑âÕπ
Y
X
D(5, 3) C(9, 3)
A(3, 2) B(7,2)
2
4
6
246810 O 2 4 6 8 10 12 14 162
4
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 190
3. °“√À¡ÿπ
°‘®°√√¡∑’Ë 3
1. „Àâπ—°‡√’¬π‡¢’¬π¿“æ∑’ˇ°‘¥®“°°“√À¡ÿπ
(1) à«π¢Õ߇ âπµ√ß AB √Õ∫®ÿ¥ O ‡ªìπ¡ÿ¡¢π“¥ 90 Õß»“ 180 Õß»“ 270 Õß»“ ·≈–
360 Õß»“ µ“¡≈”¥—∫
(2) à«π¢Õ߇ âπµ√ß AB √Õ∫®ÿ¥ O ‡ªìπ¡ÿ¡ 90 Õß»“ 180 Õß»“ 270 Õß»“ µ“¡≈”¥—∫
A BO
A BO
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1 91
(3) à«π¢Õ߇ âπµ√ß AB √Õ∫®ÿ¥ O ∑«π‡¢Á¡π“Ãî°“ ‡ªìπ¡ÿ¡¢π“¥ 90 Õß»“
(4) à«π¢Õ߇ âπµ√ß AB √Õ∫®ÿ¥ O µ“¡‡¢Á¡π“Ãî°“ ‡ªìπ¡ÿ¡¢π“¥ 90 Õß»“
(5) à«π¢Õ߇ âπµ√ß AB √Õ∫®ÿ¥ O ∑«π‡¢Á¡π“Ãî°“ ‡ªìπ¡ÿ¡¢π“¥ 180 Õß»“
A
B
O
A
B
O
A
B
O
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 192
2. °”Àπ¥„Àâ®ÿ¥°”‡π‘¥‡ªìπ®ÿ¥À¡ÿπ¢Õß°√“ø ®ß‡¢’¬π°√“ø¢Õß®ÿ¥· ¥ß°“√À¡ÿπµ“¡‡¢Á¡
π“Ãî°“ ‡ªìπ¡ÿ¡¢π“¥ 90 Õß»“
3. °”Àπ¥®ÿ¥ (2, 2) ‡ªìπ®ÿ¥À¡ÿπ¢Õß°√“ø ®ß‡¢’¬π°√“ø¢Õß®ÿ¥ (5, 4) · ¥ß°“√À¡ÿπ∑«π
‡¢Á¡π“Ãî°“ ‡ªìπ¡ÿ¡¢π“¥ 90 Õß»“
4. °”Àπ¥√Ÿª “¡‡À≈’ˬ¡ ABC ®ß √â“ß√Ÿª∑’ˇ°‘¥®“°°“√À¡ÿπ√Ÿª “¡‡À≈’ˬ¡ ABC √Õ∫®ÿ¥ O
µ“¡‡¢Á¡π“Ãî°“ ‡ªìπ¢π“¥¢Õß¡ÿ¡ 90 Õß»“ µ“¡¢—ÈπµÕπµàÕ‰ªπ’È
1. ≈“° OA
2. ≈“° OX →
∑”¡ÿ¡ 90 Õß»“ ∑‘»∑“ßµ“¡‡¢Á¡π“Ãî°“°—∫ OA √â“ß OA ‚¥¬„™â O
‡ªìπ®ÿ¥»Ÿπ¬å°≈“ß °“ß√—»¡’ OA µ—¥ OX →
∑’Ë®ÿ¥ A
3. „π∑”πÕ߇¥’¬«°—π π—°‡√’¬π “¡“√∂À“®ÿ¥À¡ÿπ¢Õß B ·≈– C ¥â«¬«‘∏’°“√‡¥’¬«°—∫®ÿ¥A ‰¥â®ÿ¥À¡ÿπ‡ªìπ B ·≈– C µ“¡≈”¥—∫
Y
X
Y
X
2
O
2 2468 4 6
2
4
2
4
O2 2
4
468 4 6
2
6
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1 93
4. ≈“° AB, BC ·≈– CA ®–‰¥â√Ÿª “¡‡À≈’ˬ¡ ABC µ“¡µâÕß°“√
5. „Àâπ—°‡√’¬π‡¢’¬π√Ÿª∑’ˉ¥â®“°°“√À¡ÿπ√Ÿª “¡‡À≈’ˬ¡ ABC √Õ∫®ÿ¥ O ‡ªìπ¢π“¥¢Õß¡ÿ¡
6. „Àâπ—°‡√’¬π‡¢’¬π√Ÿª∑’ˉ¥â®“°°“√À¡ÿπ√Ÿª “¡‡À≈’ˬ¡ ABC √Õ∫®ÿ¥ A ‡ªìπ¢π“¥¢Õß¡ÿ¡
A
B
C
O
AB
C
O
AB
C
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 194
7. „Àâπ—°‡√’¬π‡¢’¬π√Ÿª∑’ˉ¥â®“°°“√À¡ÿπ√Ÿª ’ˇÀ≈’ˬ¡ ABCD √Õ∫®ÿ¥ A ‡ªìπ¢π“¥¡ÿ¡
8. „Àâπ—°‡√’¬π‡¢’¬π√Ÿª∑’ˉ¥â®“°°“√À¡ÿπ√Ÿª‡À≈’ˬ¡ ABCD √Õ∫®ÿ¥ O ‡ªìπ¢π“¥¢Õß¡ÿ¡
9. (1) ®ß‡¢’¬π¿“殓°°“√À¡ÿπ√Ÿª “¡‡À≈’ˬ¡ ABC √Õ∫®ÿ¥ O ∑«π‡¢Á¡π“Ãî°“ ‡ªìπ¡ÿ¡¢π“¥90 Õß»“
(2) ®ß‡¢’¬π¿“殓°°“√À¡ÿπ√Ÿª “¡‡À≈’ˬ¡ ABC √Õ∫®ÿ¥ O µ“¡‡¢Á¡π“Ãî°“ ‡ªìπ¡ÿ¡¢π“¥180 Õß»“
(1)
A
B
C
D
A
B C
D
O
A
B
C
O
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1 95
(2)
10. ®ß‡¢’¬π¿“æ∑’ˇ°‘¥®“°°“√À¡ÿπ√Ÿª ’ˇÀ≈’ˬ¡ PQRS √Õ∫®ÿ¥ O ∑«π‡¢Á¡π“Ãî°“ ‡ªìπ¡ÿ¡¢π“¥180 Õß»“
A
B
C
O
O
P
Q
R
S
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 196
11. ®ß‡¢’¬π¿“æ∑’ˇ°‘¥®“°°“√À¡ÿπ√Ÿª “¡‡À≈’ˬ¡ ABC √Õ∫®ÿ¥ O ∑«π‡¢Á¡π“Ãî°“ ‡ªìπ¡ÿ¡¢π“¥60 Õß»“
12. ®ß‡¢’¬π¿“æ∑’ˇ°‘¥®“°°“√À¡ÿπ√Ÿª ’ˇÀ≈’ˬ¡ PQRS √Õ∫®ÿ¥ O µ“¡‡¢Á¡π“Ãî°“ ‡ªìπ¡ÿ¡¢π“¥120 Õß»“
O
B
C
A
S
R
PQ
O
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1 97
Y
X
R(1, 4)
Q(4, 1)
P(2, 3)
1O
2
3
4
1234
1
2
3
4
1 2 3 4
A
13. ®ß‡¢’¬π¿“殓°°“√À¡ÿπ√Ÿª “¡‡À≈’ˬ¡ PQR √Õ∫®ÿ¥ A(1, 0) ∑«π‡¢Á¡π“Ãî°“ ‡ªìπ¡ÿ¡¢π“¥ 90 Õß»“
4. °“√·ª≈ß∑“߇√¢“§≥‘µ
°‘®°√√¡∑’Ë 4
1. °”Àπ¥√Ÿª “¡‡À≈’ˬ¡ ABC ®ß‡¢’¬π°√“ø
(1) √Ÿª∑’ˇ°‘¥®“°°“√‡≈◊ËÕπ¢π“π¿“¬„µâ°“√‡≈◊ËÕπ¢π“π
2
3
(2) √Ÿª∑’ˇ°‘¥®“°°“√ –∑âÕπ¢Õß√Ÿª “¡‡À≈’ˬ¡ ABC °—∫·°π Y
(3) √Ÿª∑’ˇ°‘¥®“°°“√À¡ÿπ√Ÿª “¡‡À≈’ˬ¡ ABC µ“¡‡¢Á¡π“Ãî°“√Õ∫®ÿ¥ O ‡ªìπ¡ÿ¡¢π“¥ 90
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 198
Y
X
C(3, 4)
A(1, 0)
1 O
B(5, 2)
2345
1
2
3
4
1
2
1 2 3 4 5
Y
X1 O2345
1
2
3
4
1
2
1 2 3 4 5
3
4
5
6 7
A(1, 0)
C(3, 4)
(5, 2)B
(1)
(2)
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1 99
(3)
2. °”Àπ¥√Ÿª “¡‡À≈’ˬ¡ ABC
(1) ®ß‡¢’¬π√Ÿª∑’ˇ°‘¥®“°°“√‡≈◊ËÕπ¢π“π¢Õß√Ÿª “¡‡À≈’ˬ¡ ABC ¿“¬„µâ°“√‡≈◊ËÕπ¢π“πµ“¡‡ âπµ√ß∑’Ë°”Àπ¥·≈–√ŸªÕ¬Ÿà„π§«Õ¥√—πµå∑’Ë 3
(2) ®ß‡¢’¬π√Ÿª∑’ˇ°‘¥®“°°“√ –∑âÕπ√Ÿª “¡‡À≈’ˬ¡ ABC ¢â“¡‡ âπµ√ß∑’˺à“π®ÿ¥ (0, 0) °—∫®ÿ¥ (1, 1)
(3) ®ß‡¢’¬π√Ÿª∑’ˇ°‘¥®“°°“√À¡ÿπ√Ÿª “¡‡À≈’ˬ¡ ABC ∑«π‡¢Á¡π“Ãî°“‡ªìπ¡ÿ¡¢π“¥ 120
Õß»“
(1)
Y
X1 O2345
1
2
3
4
1
2
1 2 3 4 5 6 7
3
4
5
C(3, 4)
B(5, 2)
A(1, 0)
C(2, 4)
A(1, 1)
B(4, 2)
Y
X1 O2345
1
2
3
4
1
2
1 2 3 4 5
3
4
5
6
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1100
(2)
(3)
°‘®°√√¡‡ πÕ·π–
„Àâπ—°‡√’¬πÕÕ°·∫∫√Ÿª‡∑ ‡´≈‡≈™—π (Tessellation) ¥â«¬µπ‡Õß·≈â«≈Õ°≈ß„π°√–¥“…À≈“¬Ê √Ÿª ·≈â«π”√Ÿª∑’˵—¥ÕÕ°¡“µàÕµ“¡À≈—°°“√ –∑âÕπ °“√À¡ÿπ ·≈–°“√‡≈◊ËÕπ¢π“π ·≈–«“¥‰«â ®–‰¥â√Ÿª‡∑ ‡´≈‡≈™—πµ“¡µâÕß°“√
Y
X1 O2345
1
2
3
4
1
2
1 2 3 4 5
C(2, 4)
A(1, 1)
B(4, 2)
Y
X1 O2345
1
2
3
4
1
2
1 2 3 4 5
C(2, 4)
A(1, 1)
B(4, 2)
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1 101
„Àâπ—°‡√’¬π‡≈◊Õ°§”µÕ∫∑’Ë∂Ÿ°µâÕß
1. «—µ∂ÿ‡≈◊ËÕπ¢π“π¡’§«“¡À¡“¬µ√ß°—∫¢âÕ„¥
1. «—µ∂ÿ‡§≈◊ËÕπ∑’Ë√Õ∫®ÿ¥§ß∑’Ë®ÿ¥Àπ÷Ëß
2. «—µ∂ÿ‡§≈◊ËÕπ∑’Ë¡’∑‘»∑“ߢÕß√Ÿªæ≈‘°°≈—∫
°—∫√Ÿªµâπ·∫∫
3. «—µ∂ÿ‡§≈◊ËÕπ∑’Ë∑”„Àâ¢π“¥·≈–√Ÿª√à“ß
¢Õß√Ÿªµâπ·∫∫‡ª≈’ˬπ‰ª
4. «—µ∂ÿ‡§≈◊ËÕπ∑’Ë∑ÿ°®ÿ¥¢Õß√Ÿªµâπ·∫∫
‡§≈◊ËÕπ∑’ˉª„π∑‘»∑“߇¥’¬«°—π‡ªìπ
√–¬–∑“߇∑à“°—π
2. ¢âÕ„¥¡’‡Àµÿ°“√≥åµ√ß°—∫ ¡∫—µ‘°“√À¡ÿπ
1. °“√‡¥‘π¢÷Èπ∫—π‰¥
2. °“√‡≈àπ™‘ߙⓠ«√√§å
3. ‡ß“¢Õßµ÷° Ÿß·ÀàßÀπ÷Ëß
4. ‡√◊Õ·≈àπµ√߉ª
3. √Ÿª„π¢âÕ„¥‰¡à‰¥â‡°‘¥®“°°“√‡≈◊ËÕπ¢π“π
1.
2.
3.
4.
4. °”Àπ¥√Ÿª ’ˇÀ≈’ˬ¡ ABCD ∂⓵—¥µ“¡
√Õ¬ª√–·≈⫇≈◊ËÕπ√Ÿª “¡‡À≈’ˬ¡‰ª¥â“π
µ√ߢⓡ®–‰¥â√Ÿª„¥
1.
·∫∫∑¥ Õ∫º≈ —¡ƒ∑∏‘Ï∑“ß°“√‡√’¬πª√–®”Àπ૬
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1102
2.
3.
4.
5. √Ÿª ’ˇÀ≈’ˬ¡ ABCD ¡’®ÿ¥¬Õ¥ A(4, 1),B(3, 1), C(3, 2) ·≈– D(2, 4) ·≈–®ÿ¥‡≈◊ËÕπ¢π“π®ÿ¥ A §◊Õ (1, 2) ®ßÀ“«à“®ÿ¥æ‘°—¥ B, C ·≈– D §◊Õ¢âÕ„¥1. B(0, 1), C(5, 3), D(4, 5)
2. B(0, 1), C(6, 3), D(5, 5)
3. B(1, 1), C(5, 3), D(4, 5)
4. B(1, 1), C(6, 3), D(5, 5)
6. æ‘°—¥¢Õß®ÿ¥∑’ˇ°‘¥®“°°“√‡≈◊ËÕπ¢π“π§◊Õ¢âÕ„¥1. ∂â“√Ÿª‡≈◊ËÕπ¢π“π‰ª∑“ߢ«“ æ‘°—¥
µ—«Àπâ“®–‡æ‘Ë¡¢÷Èπ2. ∂â“√Ÿª‡≈◊ËÕπ¢π“π‰ª∑“ß â“¬ æ‘°—¥
µ—«À≈—ß®–≈¥≈ß3. ∂â“√Ÿª‡≈◊ËÕπ¢π“π≈ߥâ“π≈à“ß æ‘°—¥
µ—«Àπâ“®–≈¥≈ß4. ∂â“√Ÿª‡≈◊ËÕπ¢π“π¢÷Èπ¥â“π∫π æ‘°—¥
µ—«Àπâ“®–‡æ‘Ë¡¢÷Èπ
7. ¢âÕ„¥‡¢’¬π√Ÿª∑’ˇ°‘¥®“°°“√‡≈◊ËÕπ¢π“π
µ“¡∑‘»∑“ß·≈–√–¬–∑“ß∑’Ë°”À𥉥â∂Ÿ°µâÕß
1.
2.
3.
A B
C
A B
C
A B
C
A B
CD
D
A B
CA B
C
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1 103
4.
8. ¢âÕ„¥∫Õ°≈—°…≥–°“√‡≈◊ËÕπ¢π“π®“°
√Ÿªµâπ·∫∫ A ‰ª¬—ß√Ÿª∑’ˇ°‘¥®“°°“√
‡≈◊ËÕπ¢π“π B
1. ‡≈◊ËÕπ¢π“π‰ª∑“ߢ«“ 1 Àπ૬ ·≈–
‡≈◊ËÕπ≈ߥâ“π≈à“ß 3 Àπ૬
2. ‡≈◊ËÕπ¢π“π‰ª∑“ߴ⓬ 1 Àπ૬ ·≈–
‡≈◊ËÕπ¢÷Èπ¥â“π∫π 3 Àπ૬
3. ‡≈◊ËÕπ¢π“π‰ª∑“ߢ«“ 3 Àπ૬ ·≈–
‡≈◊ËÕπ≈ߥâ“π≈à“ß 1 Àπ૬
4. ‡≈◊ËÕπ¢π“π‰ª∑“ߴ⓬ 3 Àπ૬ ·≈–
‡≈◊ËÕπ¢÷Èπ¥â“π∫π 1 Àπ૬
9. ∂â“√Ÿª A ‡°‘¥®“°°“√‡≈◊ËÕπ¢π“π¢Õß√Ÿª B
·≈â« ¢âÕ„¥‡ªìπ°“√Õ∏‘∫“¬°“√‡≈◊ËÕπ
¢π“π®“°√Ÿª∑’Ë°”Àπ¥
1. ‡≈◊ËÕπ¢π“π‰ª∑“ߢ«“ 3 Àπ૬ ·≈–
‡≈◊ËÕπ¢÷Èπ¥â“π∫π 3 Àπ૬
2. ‡≈◊ËÕπ¢π“π‰ª∑“ߢ«“ 6 Àπ૬ ·≈–
‡≈◊ËÕπ¢÷Èπ¥â“π∫π 6 Àπ૬
3. ‡≈◊ËÕπ¢π“π‰ª∑“ߴ⓬ 3 Àπ૬ ·≈–
‡≈◊ËÕπ≈ߥâ“π≈à“ß 3 Àπ૬
4. ‡≈◊ËÕπ¢π“π‰ª∑“ߴ⓬ 6 Àπ૬ ·≈–
‡≈◊ËÕπ≈ߥâ“π≈à“ß 6 Àπ૬
10. √Ÿª„π¢âÕ„¥‡¡◊ËÕÀ¡ÿπ®ÿ¥∑’Ë°”À𥇪ìπ¡ÿ¡
180 Õß»“·≈⫉¥â√Ÿªµâπ·∫∫
1. 2.
3. 4.
A B
C
A B
CD
D
A
B
A
B
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1104
11. √Ÿª„π¢âÕ„¥µàÕ‰ªπ’ȇ°‘¥®“°°“√À¡ÿπ
1. 2.
3. 4.
12. √Ÿª„π¢âÕ„¥µàÕ‰ªπ’ȉ¡à¡’ ¡¡“µ√°—∫°“√
À¡ÿπ
1. 2.
3. 4.
13. ∂â“ P ‡ªìπ√Ÿªµâπ·∫∫ ·≈–®ÿ¥ O ‡ªìπ®ÿ¥
À¡ÿπ·≈â«√Ÿª Q ‡ªìπº≈®“°°“√À¡ÿπ√Ÿª P
„π≈—°…≥–„¥
1. À¡ÿπµ“¡‡¢Á¡π“Ãî°“ 90 Õß»“
2. À¡ÿπ∑«π‡¢Á¡π“Ãî°“ 180 Õß»“
3. À¡ÿπµ“¡‡¢Á¡π“Ãî°“ 270 Õß»“
4. À¡ÿπ∑«π‡¢Á¡π“Ãî°“ 270 Õß»“
14. ®ÿ¥„¥‡°‘¥®“°°“√À¡ÿπ®ÿ¥ A √Õ∫®ÿ¥ G
µ“¡‡¢Á¡π“Ãî°“ 90 Õß»“
G. H. I.F. E. D.A. B. C.
1. ®ÿ¥ B 2. ®ÿ¥ E
3. ®ÿ¥ D 4. ®ÿ¥ I
15. ABC ¡’®ÿ¥¬Õ¥ A(1, 4), B(3, 2) ·≈–
C(5, 3) ∂â“ ABC ‡°‘¥®“°°“√À¡ÿπ
ABC ∑«π‡¢Á¡π“Ãî°“ 270 Õß»“ √Õ∫
®ÿ¥°”‡π‘¥ ®–‰¥â®ÿ¥¬Õ¥¢Õß ABC µ√ß
°—∫¢âÕ„¥
1. A(1, 4), B(3, 2), C(5, 3)
2. A(1, 4), B(3, 2), C(5, 3)
3. A(4, 1), B(2, 3), C(3, 5)
4. A(4, 1), B(2, 3), C(3, 5)
16. √Ÿª„π¢âÕ„¥‡°‘¥®“°°“√ –∑âÕπ√Ÿª A ‡æ’¬ß
Õ¬à“߇¥’¬«
1. 2.
3. 4.
P
Q
O
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1 105
17. √Ÿª„π¢âÕ„¥¡’ ¡¡“µ√°“√ –∑âÕπ
1.
2.
3.
4.
18. √Ÿª„π¢âÕ„¥∑’ˇ°‘¥®“°°“√ –∑âÕπ‰¥â∂Ÿ°µâÕß
1.
2.
3.
4.
19. √Ÿª„π¢âÕ„¥‡°‘¥®“°°“√ –∑âÕπ√Ÿª A ∫π
‡ âπ
1. √Ÿª B
2. √Ÿª B ·≈–√Ÿª C3. √Ÿª E4. √Ÿª E ·≈–√Ÿª D
20. ∂â“ –∑âÕπ√Ÿª “¡‡À≈’ˬ¡¡ÿ¡©“° ‚¥¬„™â¥â“πµ√ߢⓡ¡ÿ¡©“°‡ªìπ‡ âπ –∑âÕπ √Ÿª∑’ˇ°‘¥¢÷Èπ®–‡ªìπ√Ÿª‡À≈’ˬ¡™π‘¥„¥1. √Ÿª ’ˇÀ≈’ˬ¡®—µÿ√— 2. √Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“3. √Ÿª ’ˇÀ≈’ˬ¡«à“«
4. ¢âÕ 1 ·≈–¢âÕ 3 ∂Ÿ°µâÕß
k
A
B
C
D
E
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1106
21. √Ÿª A ‡°‘¥®“°°“√ –∑âÕπ√Ÿª B ¥â«¬‡ âπ
–∑âÕπ‡ âπ„¥„π√Ÿª∑’Ë°”Àπ¥µàÕ‰ªπ’È
1. ‡ âπ X 2. ‡ âπ X
3. ‡ âπ Y 4. ‡ âπ Y
®ß„™â¢âÕ¡Ÿ≈µàÕ‰ªπ’ȵÕ∫§”∂“¡¢âÕ 22-23
√Ÿª ’Ë ‡À≈’ˬ¡ ABCD ¡’®ÿ¥¬Õ¥ A(3, 2),
B(7, 2), C(1, 5) ·≈– D(5, 5)
22. æ‘°—¥®ÿ¥¬Õ¥¢Õß√Ÿª ’ˇÀ≈’ˬ¡ ABCD ∑’Ë
‡°‘¥®“°°“√ –∑âÕπ√Ÿª ’ˇÀ≈’ˬ¡ ABCD
¢â“¡·°π Y §◊Õ¢âÕ„¥
1. A(3, 2), B(7, 2), C(1, 5), D(5, 5)
2. A(3, 2), B(7, 2), C(1, 5),
D(5, 5)
3. A(3, 2), B(7, 2), C(1, 5),
D(5, 5)
4. A(2, 3), B(2, 7), C(5, 1),
D(5, 5)
23. æ‘°—¥®ÿ¥¬Õ¥¢Õß√Ÿª ’ˇÀ≈’ˬ¡ ABCD ∑’Ë
‡°‘¥®“°°“√ –∑âÕπ√Ÿª ’ˇÀ≈’ˬ¡ ABCD
¢â“¡·°π X §◊Õ¢âÕ„¥
1. A(3, 2), B(7, 2), C(1, 5), D(5, 5)
2. A(3, 2), B(7, 2), C(1, 5),
D(5, 5)
3. A(3, 2), B(7, 2), C(1, 5),
D(5, 5)
4. A(2, 3), B(2, 7), C(5, 1),
D(5, 5)
24. √Ÿª∑’Ë°”Àπ¥„ÀâµàÕ‰ªπ’ȇ°‘¥®“°°“√·ª≈ß
·∫∫„¥
1. °“√À¡ÿπ
2. °“√ –∑âÕπ
3. °“√‡≈◊ËÕπ¢π“π
4. °“√ –∑âÕπ¢â“¡‡ âπ –∑âÕπ 2 ‡ âπ∑’Ë
¢π“π°—π
25. √Ÿª∑’ˇ°‘¥®“°°“√ –∑âÕπ¢â“¡‡ âπ –∑âÕπ 2
‡ âπ∑’Ë¢π“π°—π ®–‡À¡◊Õπ°—∫√Ÿª∑’ˇ°‘¥
®“°°“√·ª≈ß·∫∫„¥
1. °“√À¡ÿπ
2. °“√‡≈◊ËÕπ¢π“π
3. °“√ –∑âÕπ·∫∫‡≈◊ËÕπ
4. °“√ –∑âÕπ¢â“¡‡ âπ –∑âÕπ 2 ‡ âπ∑’Ë
µ—¥°—π
A
BC
D
Y Y
X
X
·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 1 107
26. √Ÿª A ‡°‘¥®“°°“√·ª≈ß√Ÿª B ‚¥¬„™â°“√
·ª≈ß·∫∫„¥®“°√Ÿª∑’Ë°”Àπ¥
1. °“√ –∑âÕπ 2 §√—Èß
2. °“√ –∑âÕπ·∫∫‡≈◊ËÕπ
3. °“√À¡ÿπ·≈–°“√‡≈◊ËÕπ
4. °“√ –∑âÕπ·≈–°“√À¡ÿπ
27. ∂â“√Ÿª ABC ‡°‘¥®“°°“√·ª≈ß√Ÿª ABC
‚¥¬∑’Ë®ÿ¥ A ·ª≈߉ª‡ªìπ®ÿ¥ A ®ÿ¥ B ·ª≈ß
‰ª‡ªìπ®ÿ¥ B ·≈–®ÿ¥ C ·ª≈߉ª‡ªìπ®ÿ¥
C ¥—ß√Ÿª
°“√·ª≈ߥ—ß°≈à“«‡ªìπ°“√·ª≈ß·∫∫„¥
1. °“√À¡ÿπ
2. °“√ –∑âÕπ
3. °“√‡≈◊ËÕπ¢π“π
4. °“√ –∑âÕπ·∫∫‡≈◊ËÕπ
28. √Ÿª„π¢âÕ„¥∑’ˇ°‘¥®“°°“√ –∑âÕπ∫π‡ âπ
–∑âÕπ Õ߇ âπ∑’Ë¢π“π°—π
1. 2.
3. 4.
29. °“√À“æ◊Èπ∑’Ë„π¢âÕ„¥∑’˵âÕßÕ“»—¬§«“¡√Ÿâ
‡√◊ËÕß°“√À¡ÿπ
1.
2.
3.
4.
30. æ◊Èπ∑’Ë à«π∑’Ë·√‡ß“¡’§à“‡∑à“°—∫¢âÕ„¥
1. 2 µ“√“ßÀπ૬ 2. 4 µ“√“ßÀπ૬
3. 6 µ“√“ßÀπ૬ 4. 8 µ“√“ßÀπ૬
A
B
A B
C
A B
C
1
2
1
1
1
2
2 2
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