เอกสารประกอบการสอนเอกสารประกอบการสอนวชิ าคณติ ศาสตรพ2 ้นื ฐาน ม.3 เทอม 1 1
วิชาคณติ ศาสตรพ3 ืน้ ฐาน
ม.3 เทอม 1
ชือ่ …………………………………………………………………………ชน้ั ………………….เลขที่ ………….
โรงเรยี น …………………………………………… ครูผ7ูสอน ………………………………………………….
เอกสารประกอบการสอนวชิ าคณิตศาสตรพ2 ื้นฐาน ม.3 เทอม 1 2
แบบฝก< หัด 1.1 เร่ือง อสมการเชิงเส7นตวั แปรเดยี ว
1. จงเขียนประโยคท่ใี ช1สัญลกั ษณท9 างคณติ ศาสตรแ9 ทนแต?ละขอ1 ความตอ? ไปน้ี (ให1 x แทนจำนวนจำนวนหนง่ึ )
1) สามเท?าของจำนวนจำนวนหน่งึ ไม?เกนิ 18
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2) ผลบวกของจำนวนจำนวนหน่ึงกบั 7 ไม?ถงึ 25
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3) ผลบวกของสามเทา? ของจำนวนจำนวนหนงึ่ กับ 6 ไมน? 1อยกวา? 20
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4) เศษเจ็ดส?วนสิบหา1 ของจำนวนจำนวนหนงึ่ ไม?เท?ากับ 105
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5) เศษสามสว? นสข่ี องผลตา? งของจำนวนจำนวนหน่งึ กบั 2 มากกว?า 40
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6) หา1 เทา? ของผลบวกของจำนวนจำนวนหนึ่งกบั 6 นอ1 ยกวา? 30
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7) ผลบวกของสามในสีข่ องจำนวนจำนวนหนง่ึ กับ 8 ไมเ? กิน 15
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8) สองเทา? ของผลตา? งของจำนวนจำนวนหนึง่ กับ 4 นอ1 ยกวา? ห1าเทา? ของผลบวกของจำนวนจำนวนนั้นกบั 8
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เอกสารประกอบการสอนวชิ าคณติ ศาสตรพ2 ืน้ ฐาน ม.3 เทอม 1 3
2. ในการใช1ถนนบางแหง? จะมปี Nายจราจรกำหนดวา? รถที่สามารถผ?านถนนเสน1 น้ีไดจ1 ะต1องมีนำ้ หนกั ของรถรวม
กบั นำ้ หนักบรรทกุ ไมเ? กนิ น้ำหนักท่ีแสดงบนปNาย
อสิ ระตอ1 งขับรถบรรทุกหกล1อทมี่ ีนำ้ หนกั ของรถ 6 ตัน ไปสง? ผลไม1ท่ตี ลาดผลไม1 โดยผ?านสะพาน
ข1ามคลองแห?งหนงึ่ ทม่ี ปี Nายกำหนดนำ้ หนกั ไวก1 อ? นขึน้ สะพาน ดงั รูป และมีผลไม1ที่ต1องนำไปสง? ตามรายการดงั น้ี
ทุเรยี นหมอนทอง 3 ตนั 300 กิโลกรัม
เงาะโรงเรยี น 1 ตนั 800 กโิ ลกรัม
กระท1อนปยุ ฝNาย 2 ตัน 500 กโิ ลกรมั
ลนิ้ จจ่ี ักรพรรด ิ 1 ตนั 700 กิโลกรมั
ขนุนไพศาลทกั ษิณ 2 ตนั 200 กโิ ลกรัม
มะม?วงโชคอนันต9 1 ตัน 800 กโิ ลกรัม
ลำไยสีชมพู 2 ตัน 400 กิโลกรมั
ส1มโอขาวแตงกวา 1 ตัน 750 กโิ ลกรมั
อิสระจะตอ1 งขบั รถบรรทุกหกล1อคนั นี้ไปสง? ผลไม1อย?างนอ1 ยกี่รอบ แตล? ะรอบบรรทุกผลไมช1 นิดใดบ1าง
และคดิ เปSนนำ้ หนกั เทา? ไร โดยแต?ละรอบจะตอ1 งบรรทกุ ผลไม1ชนดิ เดียวกนั ไปพรอ1 มกนั ทั้งหมด
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เอกสารประกอบการสอนวิชาคณิตศาสตรพ2 ืน้ ฐาน ม.3 เทอม 1 4
แบบฝก< หดั 1.2 เรื่อง คำตอบของอสมการเชงิ เสน7 ตวั แปรเดยี ว
1. จงเขยี นกราฟแสดงคำตอบของอสมการในแต?ละข1อตอ? ไปนี้
1) x ≥ 18 2) a < −3
3) n ≤ 12 4) x ≠ 20
5) −9 < x < 9 6) 18 ≤ x < 24
2. จงหาว?ากราฟแสดงคำตอบในแต?ละข1อต?อไปน้ีแสดงจำนวนใดบ1าง
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เอกสารประกอบการสอนวิชาคณิตศาสตรพ2 น้ื ฐาน ม.3 เทอม 1 5
3. กำหนดให 1 x ≥ 20 ! จงหาค?า x ท่ีน1อยท่สี ดุ
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1) เมื่อ x เปนS จำนวนเตม็ 2) เมอ่ื x เปSนจำนวนค?ู
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3) เมอื่ x เปSนจำนวนเฉพาะ 4) เม่อื x เปนS จำนวนคี่
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5) เม่ือ x เปนS จำนวนจรงิ
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4. กำหนดให1 −10 ≤ x < 10
1) จงหาค?า x ที่นอ1 ยท่สี ดุ เม่อื x เปนS จำนวนเตม็
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2) จงหาคา? x ท่ีมากท่สี ดุ เมอ่ื x เปSนจำนวนเตม็
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3) จงหาคา? x ทั้งหมดที่สอดคล1องกบั อสมการทก่ี ำหนดให1 เมื่อ x เปSนจำนวนเตม็
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เอกสารประกอบการสอนวชิ าคณิตศาสตรพ2 ืน้ ฐาน ม.3 เทอม 1 6
แบบฝ<กหดั 1.3 เร่ือง การแก7อสมการเชิงเสน7 ตัวแปรเดยี ว
1. จงแกอ1 สมการตอ? ไปน้ี และเขยี นกราฟแสดงคำตอบ
1) x + 5 ≥ 12 2) 10 − a > 8
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3) !# − 1 ≤ 5 "
4) − + 2 < −3
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5) %$ + 2 ≠ 3 6) 3(m − 1) < 18
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7) $&!' ≥ 7 8) $&(' ≠ 14
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9) b + 4 < 2b − 1 10) 0.7a ≠ 0.3a + 2
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11) 4x − 3 < 6x + 6 12) 5n − 3 > n + 2
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เอกสารประกอบการสอนวิชาคณิตศาสตรพ2 ืน้ ฐาน ม.3 เทอม 1 7
2. จงแกอ1 สมการตอ? ไปน ี้ 2) −2x − 10 > 5x + 4
1) 2x − 7 > 45
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! 4) (" "
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3) − x ≥ x + 5 x + 2 ≠ x + 5
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6) 2(x − 15) ≤ −(3x + 5)
5) 24x − 4.5 < 15(x + 0.3)
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8) !" (2x + 1) ≤ ' (10x − 3)
7) $&%' ≥ 8 − 2x )
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9) 3 ;* !, -< ≤ −2 ;*"&)< 10) 2x + 3x ≠ 4x + 5x
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เอกสารประกอบการสอนวชิ าคณติ ศาสตรพ2 ืน้ ฐาน ม.3 เทอม 1 8
แบบฝก< หัด 1.4 เร่อื ง โจทยHปIญหาเก่ยี วกับอสมการเชิงเส7นตัวแปรเดยี ว
วันแรกอ?านได1 !
1. มาวินอ?านหนังสอื เลม? หนึ่ง ของเล?ม วนั ต?อมาอา? นไดอ1 กี 25 หน1า รวมสองวนั
.
อา? นได1มากกว?าครง่ึ เล?ม จงหาวา? หนงั สือเลม? นีม้ ีจำนวนหน1าอยา? งมากก่หี น1า
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2. ปWญญามีเหรยี ญบาทและเหรียญหา1 บาทอยู?ในกระปXองออมสินจำนวนหนึ่ง เมอ่ื เหรียญเต็มกระปอX ง
เขาเทออกมานับ พบว?ามีเหรียญบาทมากกวา? เหรยี ญหา1 บาทอยู? 12 เหรียญ และนับเปSนเงินได1ไมน? อ1 ยกว?า
300 บาท จงหาว?ามเี หรยี ญห1าบาทอยูอ? ย?างน1อยก่ีเหรียญ
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3. ถา1 สองเทา? ของจำนวนเต็มบวกจำนวนหนึง่ มากกวา? 20 อย?ไู มถ? ึง 6 จำนวนดังกลา? วเปนS จำนวนใดได1บา1 ง
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เอกสารประกอบการสอนวชิ าคณิตศาสตรพ2 นื้ ฐาน ม.3 เทอม 1 9
4. รูปสีเ่ หลยี่ มผนื ผ1ารูปหน่ึงมีอตั ราสว? นของความกว1างต?อความยาวเปนS 3 : 5 และมีความยาวรอบรูปไม?นอ1 ยกว?า
48 เซนติเมตร รูปสเ่ี หลย่ี มผนื ผา1 รูปนมี้ ีพ้ืนทอี่ ยา? งนอ1 ยเทา? ไร
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5. จำนวนจำนวนหนึง่ รวมกบั "' ของผลบวกของจำนวนนั้นกบั 5 แลว1 ยงั น1อยกวา? 7 จงหาวา? จำนวนดงั กลา? ว
เปนS จำนวนใดได1บา1 ง ถา1
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1) จำนวนนั้นเปSนจำนวนนบั
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2) จำนวนน้นั เปSนจำนวนเตม็
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6. อาทติ ยส9 อบไดค1 ะแนนวิชาวทิ ยาศาสตรค9 ร้ังท่ี 1 ครงั้ ท่ี 2 และครงั้ ที่ 3 เปSน 69, 75 และ 72 คะแนน
ตามลำดับ จากคะแนนเตม็ 100 คะแนน จงหาว?า อาทิตยจ9 ะตอ1 งได1คะแนนสอบคร้ังที่ 4 อย?างน1อยเท?าไร
จากคะแนนเตม็ 100 คะแนน จึงจะทำให1คะแนนเฉล่ยี ของการสอบท้งั สี่ครัง้ เปนS 75 คะแนน ขึ้นไป
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เอกสารประกอบการสอนวิชาคณิตศาสตรพ2 ้ืนฐาน ม.3 เทอม 1 10
7. แม?คา1 ตอ1 งการบรรจุมะมว? งใส?ลังไม1 ซึ่งลังเปลา? แต?ละใบหนัก
2.5 กิโลกรัม มะม?วงแต?ละผลมีขนาดใกล1เคียงกัน หนกั ผลละ
0.3 กโิ ลกรัม เพอ่ื เปนS การประหยัดค?าใชจ1 ?ายในการขนสง? จึงตอ1 ง
บรรจมุ ะมว? งให1ไดม1 ากท่สี ุด แต?ต1องไม?หนกั เกนิ ไปจนเปSนปญW หา
ในการเคล่อื นย1าย จากประสบการณแ9 มค? า1 พบวา? ถา1 จะใหค1 1ุม
คา? ขนสง? โดยมะมว? งไม?เสยี หาย ต1องบรรจมุ ะม?วงให1แต?ละลังมี
น้ำหนักของมะม?วงรวมกบั น้ำหนักของลงั แล1วไมน? อ1 ยกว?า 19 กิโลกรมั แต?ไม?เกนิ 25 กิโลกรัม
จงหาว?าแม?คา1 ควรบรรจุมะมว? งใส?ลงั อยา? งนอ1 ยลงั ละกผี่ ล และอยา? งมากลังละกผี่ ล
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8. พัดชาซอ้ื เหมาไข?ไก?มา 1,000 ฟอง เปSนเงนิ 3,600 บาท โดยมไี ขไ? กเ? บอร9 0 และเบอร9 1 คละกัน
แล1วนำมาขายปลีกโดยขายไข?ไก?เบอร9 0 ฟองละ 4 บาท และไขไ? ก? เบอร9 1 ฟองละ 3.50 บาท จงหาว?า
1) ตอ1 งมีไข?ไกเ? บอร9 0 อย?างน1อยก่ฟี อง พัดชาจึงจะได1กำไร เมือ่ ขายไขไ? กไ? ดท1 ั้งหมด
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2) ตอ1 งมีไข?ไก?เบอร9 1 ไม?เกินก่ฟี อง พดั ชาจึงจะไดก1 ำไรมากกว?า 325 บาท เมอื่ ขายไข?ไก?ไดท1 ้งั หมด
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เอกสารประกอบการสอนวิชาคณิตศาสตรพ2 นื้ ฐาน ม.3 เทอม 1 11
9. นพิ ิธไดร1 ับเงินจากพ?อเพื่อเปนS คา? ใช1จ?ายประจำสปั ดาห9 เมอื่ สนิ้ สัปดาหเ9 ขารวมรายจ?ายแล1วพบวา? ได1ใชเ1 งินไป
ทงั้ สน้ิ 420 บาท และฝากธนาคารไว1 100 บาท เมือ่ นับเงินทเี่ หลอื แลว1 ปรากฏว?าเหลอื ไม?ถึง 50 บาท
จงหาวา? นิพธิ ไดร1 ับเงินจากพอ? เท?าไร
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10. จำนวนนกั เรยี นหอ1 ง ม. 3/1 นอ1 ยกว?าจำนวนนักเรียนห1อง ม. 3/2 อยู? 6 คน และ "! ของจำนวนนกั เรียน
มากกวา? '
ห1อง ม. 3/1 ! ของจำนวนนกั เรยี นห1อง ม. 3/2 อย?ูไม?เกิน 5 คน จงหาว?าจำนวนนกั เรียนหอ1 ง ม. 3/1
เปSนจำนวนใดไดบ1 า1 ง
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เอกสารประกอบการสอนวิชาคณิตศาสตรพ2 ืน้ ฐาน ม.3 เทอม 1 12
แบบฝ<กหดั ทา7 ยบท เรือ่ ง อสมการเชงิ เส7นตวั แปรเดียว
1. จงแกอ1 สมการตอ? ไปน้ี และเขียนกราฟแสดงคำตอบ
1) 3x + 1 ≥ 7 + 2x 2) "$ + 1 ≤ 3x + 4
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3) 10x − 9 ≥ 12x − 14 4) 3 − 3(5 − 2x) > 0
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5) 11(x − 2) < −x + 2 x − 5 ≠ x + 2
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7) !" (16 − 5x) < 2x 8) 7(x + 3) − 5 ≤ 4x + 8
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9) "! (4x − 3) ≠ 17x + 2 10) 5 ;4x + ".< ≥ 17(x − 2) + 4
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11) .($ ," '') < 2(x − 4) + 1 12) "$!&! − 7 ≠ x − 8
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เอกสารประกอบการสอนวิชาคณติ ศาสตรพ2 ้ืนฐาน ม.3 เทอม 1 13
2. จำนวนเตม็ บวกสองจำนวนตา? งกนั อยู? 8 ถา1 นำ 3 เท?าของจำนวนนอ1 ยบวกกบั จำนวนมาก จะไดผ1 ลบวก
มากกวา? 48 แตไ? มเ? กนิ 68 จำนวนเตม็ บวกท่เี ปSนจำนวนนอ1 ยคือจำนวนใด
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3. ชมขวัญมเี หรยี ญห1าบาทและเหรยี ญบาทในกระปุกออมสินท่เี ก็บไว1คิดเปนS เงิน 110 บาท
ปรากฏว?ามเี หรียญมากกว?า 50 เหรยี ญ แต?ไมถ? ึง 60 เหรียญ จงหาวา? เหรียญห1าบาททชี่ มขวัญเก็บไวม1 กี ่ีเหรยี ญ
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4. พละเตรยี มเงนิ ไปซื้อของขวัญวันเด็กสำหรับบุตร 2 คน โดยตง้ั ใจวา? ของขวญั สำหรับบุตรชาย
ตอ1 งมรี าคาไม?เกนิ "' ของเงนิ ทเ่ี ตรียมไป และของขวญั สำหรับบตุ รสาวจะแพงกวา? ของบตุ รชายไม?เกนิ 50 บาท
เมอื่ เขาซื้อของได1เรียบรอ1 ยแลว1 ปรากฏวา? จ?ายเงนิ ไปไมเ? กิน 500 บาท อยากทราบวา? พละเตรยี มเงินไปเท?าไร
บตุ รชายและบุตรสาวไดร1 ับของขวญั มรี าคาไม?เกินกี่บาท
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เอกสารประกอบการสอนวชิ าคณติ ศาสตรพ2 ื้นฐาน ม.3 เทอม 1 14
5. ในการประเมนิ ผลวชิ าคณิตศาสตร9 มีการสอบเกบ็ คะแนน 3 คร้งั รวมคะแนนเต็ม 200 คะแนน
โดยมีเกณฑ9การประเมินผล ดังน้ี
เกณฑก% ารประเมินผล ผลการเรยี น
คะแนนตั้งแต? 80% ขน้ึ ไป 4.00
คะแนนตั้งแต? 75% ข้นึ ไป แต?ไม?ถงึ 80% 3.50
คะแนนต้งั แต? 70% ขึ้นไป แต?ไม?ถงึ 75% 3.00
คะแนนตงั้ แต? 65% ขึ้นไป แต?ไม?ถึง 70% 2.50
คะแนนตง้ั แต? 60% ขน้ึ ไป แต?ไมถ? งึ 65% 2.00
คะแนนตง้ั แต? 55% ข้นึ ไป แตไ? ม?ถงึ 60% 1.50
คะแนนตั้งแต? 50% ขน้ึ ไป แต?ไม?ถึง 55% 1.00
คะแนนไมถ? งึ 50% 0
ในการสอบ นุม? นิ่มสอบเกบ็ คะแนนคร้ังที่ 1 และครง้ั ท่ี 2 ได1คะแนน 42 และ 40 คะแนน จากคะแนนเต็ม
50 คะแนน ตามลำดับ จากขอ1 มลู ขา1 งตน1 จงตอบคำถามตอ? ไปนี้
1) ในการสอบเกบ็ คะแนนครั้งที่ 3 มคี ะแนนเต็ม 100 คะแนน น?มุ น่ิมจะต1องทำคะแนนให1ได1
อย?างน1อยกีค่ ะแนน จึงจะไดผ1 ลการเรียนเปนS 4.00
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2) ในการประกาศผลคะแนนสอบในคร้งั ท่ี 2 ปรากฏวา? มกี ารประกาศผลคะแนนของนมุ? นิม่ สลบั กับ
เพอ่ื นทำให1คะแนนของน?มุ นิ่มลดลงจากทปี่ ระกาศ 6 คะแนน นักเรยี นคิดว?า นม?ุ น่ิมยงั สามารถทำคะแนนสอบ
ในคร้งั ท่ี 3 เพอ่ื ใหไ1 ดผ1 ลการเรียนเปนS 4.00 ไดห1 รอื ไม? ถ1าได1 นุม? นิม่ จะต1องได1คะแนนสอบคร้งั ท่ี 3
อยา? งน1อยก่ีคะแนน จากคะแนนเต็ม 100 คะแนน จงึ จะไดผ1 ลการเรียนเปSน 4.00
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เอกสารประกอบการสอนวิชาคณิตศาสตรพ2 ื้นฐาน ม.3 เทอม 1 15
6. นวัตตอ1 งการขน้ึ รถรับจา1 งเพอ่ื ไปซือ้ ของใช1ทหี่ า1 งสรรพสนิ ค1าแห?งหนึ่ง โดยรถรบั จา1 งคิดค?าบริการเร่มิ ต1นที่
40 บาท และคดิ ค?าบรกิ ารเพิม่ เตมิ อีกกิโลเมตรละ 5 บาท เมือ่ กลับมาถึงบ1าน นวตั ต1องจดบันทกึ รายรับและ
รายจา? ยประจำวันแต?ปรากฏว?า นวัตทำใบเสรจ็ รับเงินคา? บรกิ ารรถรับจา1 งหายไป แตน? วตั จำได1ว?า จ?ายค?าบริการ
รถรับจ1างไป – กลับ ระหว?างบา1 นกบั ห1างสรรพสินค1าทั้งหมดเปSนเงินมากกวา? 140 บาท แตไ? มเ? กิน 160 บาท
อยากทราบว?า นวตั ขน้ึ รถรับจา1 งไปห1างสรรสินคา1 ใด และจา? ยค?าบรกิ ารรถรบั จา1 งไปท้ังหมดกี่บาท
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เอกสารประกอบการสอนวิชาคณติ ศาสตรพ2 ้ืนฐาน ม.3 เทอม 1 16
7. ฮานะต1องการนำเชือกมาลอ1 มเปนS รูปเรขาคณติ แตเ? ชอื กยาวเพยี ง 280 เซนตเิ มตร อยากทราบวา?
1) รูปส่เี หลยี่ มจตั ุรสั ทีม่ พี น้ื ทีม่ ากท่สี ุดทีฮ่ านะสรา1 งไดจ1 ากเชือกเส1นนี้ จะมพี นื้ ทกี่ ีต่ ารางเซนตเิ มตร
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2) ฮานะจะสามารถสรา1 งรปู สีเ่ หลยี่ มมุมฉากที่มีความยาวมากกวา? ความกวา1 ง 20 เซนตเิ มตร
ได1หรอื ไม ? และรปู ส่ีเหล่ียมมุมฉากนัน้ จะมีพ้นื ที่มากทส่ี ดุ ก่ีตารางเซนติเมตร
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3) รูปวงกลมท่ีมพี ืน้ ท่ีมากทสี่ ุดทีฮ่ านะสรา1 งได1จากเชือกเสน1 น้ี จะมพี ้ืนทีท่ ีก่ ี่ตารางเซนตเิ มตร
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4) รปู สามเหลี่ยมมุมฉากหนา1 จัว่ ท่ีมีพนื้ ทม่ี ากท่ีสดุ ทฮี่ านะสรา1 งได1จากเชอื กเส1นน้ี
จะมพี ้นื ที่ก่ตี ารางเซนติเมตร
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เอกสารประกอบการสอนวชิ าคณติ ศาสตรพ2 ืน้ ฐาน ม.3 เทอม 1 17
8. เจ1าของฟารม9 เล้ียงสตั วแ9 ห?งหนงึ่ พบว?าอัตราสว? นของจำนวนเปSด ไก? และห?าน ท่เี ขาเลย้ี งเปSน 3 : 2 : 1
ต?อมาเขาขายเปSดไป 500 ตวั ขายไกไ? ป 200 ตัว เม่อื นบั จำนวนสตั วแ9 ล1วปรากฏว?าเหลือไมถ? งึ คร่ึงหนึง่
ของจำนวนสตั วท9 ั้งหมด จงหาจำนวนสัตว9เล้ียงท้ังหมดทเี่ ปSนไปได1ทีเ่ จา1 ของฟารม9 เล้ยี งไว1
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เอกสารประกอบการสอนวชิ าคณติ ศาสตรพ2 ้นื ฐาน ม.3 เทอม 1 18
แบบฝก< หดั 2.1 เร่ือง การแยกตัวประกอบของพหนุ ามที่อยใNู นรปู ผลบวก
และผลตNางของกำลงั สาม
จงแยกตวั ประกอบของพหนุ ามตอ? ไปน ี้
1. x" + 27 2. y" + 64
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3. 8x" + 1 4. 64z" + 125
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5. 27x" + 512y" 6. 729 + (x − 2)"
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7. (3x − 1)" + (x − 4)"
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8. (2x + 5)" + (5x − 9)"
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9. x" − 1 10. z" − 216
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เอกสารประกอบการสอนวิชาคณิตศาสตรพ2 ืน้ ฐาน ม.3 เทอม 1 19
11. 125y" − 64 12. 1,000 − 216x"
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13. 1,331y" − 343z"
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14. (4x + 3)" − 125
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15. 8 − (8x − 1)"
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16. (8x + 1)" + 8
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17. (5x − 2)" + (5x + 2)"
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เอกสารประกอบการสอนวชิ าคณิตศาสตรพ2 ืน้ ฐาน ม.3 เทอม 1 20
18. (2x + 5)" − (2x − 5)"
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19. (7x − 2)" − (6x + 9)"
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20. (8x − 15)" − (3x − 7)"
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21. x" − x! − x + 1
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22. y" + y! − 4y – 64
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เอกสารประกอบการสอนวชิ าคณิตศาสตรพ2 ้ืนฐาน ม.3 เทอม 1 21
แบบฝก< หัด 2.2 เรอื่ ง การแยกตวั ประกอบของพหุนามท่มี ดี ีกรีสูงกวNาสาม
จงแยกตวั ประกอบของพหนุ ามตอ? ไปน ี้
1. x% + 3x! 2. 4x% − 16x!
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3. x) − 81x% 4. x) + 2x% + x!
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5. x% − 625 6. 81y% − 625
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7. 160,000x% − 10,000 8. x% − 50x! + 625
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9. 25y% + 120y! + 144 10. 77x% + 170x! + 77
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11. 77x% − 72x! − 77 12. x% − 10x! + 9
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13. x% − 29x! + 100 14. 64x) − 729
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เอกสารประกอบการสอนวชิ าคณติ ศาสตรพ2 ้ืนฐาน ม.3 เทอม 1 22
15. x) + 216 16. 512 − y)
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………………………………………………………………............................……. 18. a)b" − 1
17. x% − y% ………………………………………………………………............................…….
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19. x) − y) ………………………………………………………………............................…….
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เอกสารประกอบการสอนวชิ าคณิตศาสตรพ2 ้นื ฐาน ม.3 เทอม 1 23
แบบฝ<กหัดทา7 ยบท เรอ่ื ง การแยกตวั ประกอบของพหนุ ามท่มี ีดีกรสี ูงกวาN สอง
1. จงแยกตวั ประกอบของพหนุ ามต?อไปน ี้
1) 8 − x" 2) 27x" + 125
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4) 1,000 − (x + 10)"
3) (x − 3)" + 27
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5) (x − 11)" − (x + 11)"
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6) (x − 5)" − (7x + 4)"
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8) x% − 17x! + 16
7) 81x% − 2,401
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10) 10x% − x! − 3
9) x% − 32x! + 256
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เอกสารประกอบการสอนวิชาคณติ ศาสตรพ2 น้ื ฐาน ม.3 เทอม 1 24
11) 36x% − 97x! + 36
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12) y) − 4,096
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13) 1 − a)
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14) 1,000,000z) − 1
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15) x) + 6x% + 9x! 16) 25x) + 10x" + 1
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17) 81x% − 256y% 18) 64 − x"y"
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19) 343x) + 1,000z) 20) x) − 4x"y + 4y!
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เอกสารประกอบการสอนวชิ าคณติ ศาสตรพ2 ื้นฐาน ม.3 เทอม 1 25
2. ถังนำ้ ทรงลกู บาศก9 A มคี วามยาวภายในด1านละ x + 4 นิ้ว และถงั น้ำทรงลกู บาศก9 B มคี วามยาวภายใน
ด1านละ x − 2 น้ิว ถงั นำ้ ทัง้ สองมนี ำ้ บรรจอุ ย?ูเตม็ อยากทราบวา? จะต1องใช1ถังน้ำ C ซง่ึ มีความจุ x! + 2x + 28
ลกู บาศกน9 ิว้ กใี่ บ จึงจะใสน? ้ำทง้ั หมดจากถังนำ้ A และถังน้ำ B ได1พอดี
(เมอ่ื x เปนS จำนวนเตม็ บวกที่มากกว?าสอง)
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3. A, B และ C เปSนจำนวนเตม็ บวก ที่ B มากกว?า A และ C มากกวา? B อย?ูเทา? ๆ กนั
ถ1าผลคูณของทั้งสามจำนวนเปนS x" + 12x! + 32x เมือ่ x เปนS จำนวนเตม็ บวก อยากทราบว?า
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1) B มากกว?า A อย?เู ท?าใด
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2) ถา1 ให1จำนวนท่ีนอ1 ยทส่ี ดุ ในสามจำนวนนี้ เทา? กับจำนวนเฉพาะสองหลกั ท่นี 1อยทีส่ ดุ
ผลคณู และผลบวกของ A, B และ C จะเปนS เท?าไร
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เอกสารประกอบการสอนวชิ าคณติ ศาสตรพ2 ้นื ฐาน ม.3 เทอม 1 26
แบบฝ<กหัด 3.1 เรือ่ ง สมการกำลังสองตวั แปรเดยี ว
1. สมการทีก่ ำหนดใหต1 ?อไปนี้ เปSนสมการกำลังสองตัวแปรเดียวหรอื ไม?
จงเขียน ลงในชอ? งท่ีเปนS คำตอบทถ่ี กู ตอ1 ง
สมการ สมการกำลังสองตัวแปรเดียว
ข2อที่ เป>น ไม@เปน>
1) −z! + 15 − 4z = 0
2) 0 = 6t! − 5
3) 4 − 3x = 0
4) − 1 u + 7 − 3 u! = 0
2 4
5) 0 = s(s − 9)
6) m! + 2n − 1 = 0
7) 0 = x + 2x
8) 12y! = 0
9) 3w! + 8w − 5 − 3w! = 0
10) 0 = 11v − 12v! + 13
2. จงเขียนสมการกำลงั สองตัวแปรเดียวต?อไปนใี้ ห1อยู?ในรูปท่วั ไป ax! + bx + c = 0 เมื่อ x เปSนตวั แปร
a, b และ c เปนS ค?าคงตวั โดยที่ a ≠ 0 พร1อมทงั้ บอกค?า a, b และ c ในแต?ละสมการ
รูปท่วั ไป a b c
ข2อท่ี สมการกำลังสองตัวแปรเดยี ว
1) x! − 3x = 8
2) 2m! − 7 = m
3) z! = −4
4) 0.8x − 1 = 1.5x!
5) (k − 1)! = 0
6) 3z(1.2 − z) = 2.4
7) y! + 6y = 2 − 3y!
8) (n + 4)! = 1 + 8n
9) (m − 9)(m + 9) = 9 − m!
10) 2 − 5k = k3!
เอกสารประกอบการสอนวิชาคณติ ศาสตรพ2 ื้นฐาน ม.3 เทอม 1 27
แบบฝก< หดั 3.2ก เรือ่ ง การแกส7 มการกำลงั สองตัวแปรเดยี ว
1. จงตรวจสอบว?า จำนวนทก่ี ำหนดให1ในชอ? งทา1 ยข1อ แต?ละจำนวนเปนS คำตอบของสมการในแตล? ะขอ1 หรอื ไม?
1) x! − 5x − 14 = 0 7 กับ -2
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2) n! − 6n + 9 = 0 3 กับ 0
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3) 2x! − 7x = 0 7 กบั 0
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4) 2y! = 15 + 7y 5 กบั − "!
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5) 0 = 25 + x! 5 กับ -5
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' กับ
!
6) 4z + 4z! = −1 − -1
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7) m! − 6 = m -2 กับ -3
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8) y! = 10y − 25 5 กบั 2
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2. จงหาคำตอบของสมการตอ? ไปน ี้
1) x! − 16 = 0 2) y! − 5y = 0
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3) z! + 3 = 0 4) x! − 1 = 0
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5) k! + k − 2 = 0 6) u! − 8u + 16 = 0
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เอกสารประกอบการสอนวชิ าคณิตศาสตรพ2 ืน้ ฐาน ม.3 เทอม 1 28
แบบฝก< หดั 3.2ข เรอ่ื ง การแก7สมการกำลังสองตวั แปรเดียวโดยวิธีการแยกตัวประกอบ
1. จงแก1สมการต?อไปน ้ี
1) x! − 225 = 0 2) (y − 21)! = 0
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3) (m + 12)! = 0 4) p! + 49 = 14p
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5) z! + 4z = −4 6) x! − 3x − 10 = 0
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7) y! + 5y + 6 = 0 8) −z! + 8z = 7
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9) −n! = 3 − 4n 10) u! + 7u + 12 = 0
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เอกสารประกอบการสอนวชิ าคณิตศาสตรพ2 ้นื ฐาน ม.3 เทอม 1 29
11) 14 − 5m = m! 12) x! = 5x
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13) p! − 7p + 12 = 0 14) 7n + 18 = n!
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15) w! = −6w 16) y! = 10y − 25
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17) z! + z = 30 18) −m! − 15m = 36
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19) p! = −8p − 16 20) t! − 100 = 21t
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เอกสารประกอบการสอนวชิ าคณิตศาสตรพ2 ื้นฐาน ม.3 เทอม 1 30
2. จงแกส1 มการตอ? ไปน ้ี 2) 3m! + 2m − 8 = 0
1) 2x! − 5x − 3 = 0
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3) 3z! + 10z + 3 = 0 ………………………………………………………………............................…….
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6) 3r! − 5r + 2 = 0
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5) 6y! + 5y − 4 = 0 ………………………………………………………………............................…….
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7) 2u! − 9u − 18 = 0 ………………………………………………………………............................…….
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9) 3n! − 2n = 5
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เอกสารประกอบการสอนวชิ าคณิตศาสตรพ2 ้นื ฐาน ม.3 เทอม 1 31
11) (m − 3)! = 16 12) (t + 12)t = −32
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13) 18p − 8 = −35p! 14) 1.4z! + 3.1z = 1
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15) 1! = r + 4 16) ' y! = 7y − 12
2!
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17) 2w ! − w = 7 18) (4n − 3)! = 49
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19) t! − 3t = 4t! − 36 20) (2m − 3)! − (m + 2)! = 0
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เอกสารประกอบการสอนวชิ าคณติ ศาสตรพ2 ้นื ฐาน ม.3 เทอม 1 32
3. ใหผ1 ลคณู ของจำนวนเต็มสองจำนวนที่อยต?ู ดิ กนั เทา? กบั 210 จงหาจำนวนเต็มทั้งสองนนั้
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4. ให1ผลคณู ของจำนวนค?จู ำนวนหนงึ่ กับจำนวนคูอ? กี จำนวนหนึ่งทีอ่ ยถ?ู ัดไปเท?ากบั 840 จงหาจำนวนค?ูสองจำนวนน้ัน
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5. ใหผ1 ลคูณของจำนวนค่ีบวกจำนวนหน่ึงกับจำนวนคบ่ี วกอกี จำนวนหนึ่งทอ่ี ยูถ? ดั ไปเปSน 675 จงหาจำนวนคีบ่ วก
สองจำนวนน้ัน
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6. ให1ผลคณู ของจำนวนจำนวนหนงึ่ กับจำนวนทนี่ อ1 ยกว?า 2 เทา? ของจำนวนนน้ั อยู? 1 เท?ากับ 3 จงหาจำนวนทง้ั
สองน้ัน
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เอกสารประกอบการสอนวิชาคณติ ศาสตรพ2 น้ื ฐาน ม.3 เทอม 1 33
7. ถา1 จำนวนจำนวนหน่ึงกบั อีกจำนวนหนึ่งซ่งึ น1อยกวา? จำนวนแรกอยู? 13 คูณกัน แลว1 ผลคณู เทา? กบั -40
จงหาจำนวนทัง้ สองนนั้
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8. ถา1 จำนวนสองจำนวนรวมกนั เทา? กบั 22 และกำลังสองของแต?ละจำนวนรวมกันเทา? กบั 274
จงหาจำนวนทัง้ สองนั้น
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9. ถ1าจำนวนสองจำนวนทตี่ ?างกันอยู? 3 และผลบวกของกำลังสองของท้งั สองจำนวนเปนS 117 จงหาจำนวน
ทั้งสองนน้ั
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10. ถา1 40 เท?าของผลบวกของจำนวนคจู? ำนวนหนงึ่ กบั จำนวนคอ?ู ีกจำนวนหนึง่ ท่ีอย?ูถดั ไป
เทา? กับ 9 เทา? ของผลคณู ของท้งั สองจำนวน จงหาจำนวนค?สู องจำนวนนน้ั
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เอกสารประกอบการสอนวิชาคณติ ศาสตรพ2 ้ืนฐาน ม.3 เทอม 1 34
แบบฝก< หดั 3.2ค เรอ่ื ง การแก7สมการกำลงั สองตัวแปรเดียวโดยการใชส7 ตู ร
1. จงแกส1 มการตอ? ไปน้ี โดยใช1ความรูเ1 ร่อื งกำลงั สองสมบูรณแ9 ละผลต?างของกำลังสอง
1) x! + 26x + 165 = 0 2) y! − 98y + 2,376 = 0
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3) z! + 756 = 60z 4) 400 + 50x = −x!
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5) 4y! − 3 = 8y + 9 6) 9z! + 6z − 8 = 0
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เอกสารประกอบการสอนวชิ าคณติ ศาสตรพ2 น้ื ฐาน ม.3 เทอม 1 35
2. จงแกส1 มการกำลงั สองตัวแปรเดยี วต?อไปน้ี โดยการใช1สูตร
1) x! − 12x + 11 = 0 2) y! − 3y − 10 = 0
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3) z! + 4z + 1 = 0 4) 3x! − 2x = −2
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5) 2x! = 3x + 14 6) 10z! = 17z − 3
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เอกสารประกอบการสอนวิชาคณิตศาสตรพ2 ืน้ ฐาน ม.3 เทอม 1 36
7) 14y = 1 + 49y! 8) 2x(x − 3) = 4(10 − x)
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9) ;2z + '!< z = z 10) y! + 3 = 1 ' y
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11) 2x! − 8x + 3 = 0 12) 3z! + 7z − 1 = 0
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13) 4y! − 4y − 35 = 0 14) 16x! − 8x + 1 = 0
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15) 21z! + 9z + 100 = 0 16) 4y! + 68y + 289 = 0
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17) !' (2x − 5)! = 2(10 − x) '-
18) − " (z! − 1) = 8z − 1
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19) 2y + (y − 2)! = 1 20) x(3 + 5x) = 20 − 7x
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เอกสารประกอบการสอนวิชาคณติ ศาสตรพ2 น้ื ฐาน ม.3 เทอม 1 37
แบบฝ<กหดั 3.3 เรอ่ื ง โจทยHปญI หาเกี่ยวกับสมการกำลงั สองตัวแปรเดยี ว
1. รปู สีเ่ หลย่ี มมมุ ฉากรปู หน่ึง มดี 1านยาวยาวกว?า 3 เทา? ของด1านกวา1 งอยู? 5 นิ้ว และมพี น้ื ที่ 138 ตารางน้วิ
จงหาความยาวของด1านแต?ละด1านของรูปสเ่ี หล่ยี มมมุ ฉากนี้
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2. ∆RST มพี น้ื ท่ี 52 ตารางเซนติเมตร มคี วามสูงนอ1 ยกวา? สองเท?าของความยาวของฐาน ST อย?ู 3 เซนติเมตร
จงหาความยาวของฐาน ST
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3. กำหนดรูปสามเหล่ียม ABC ท่มี ี ABVC เปนS มุมฉาก WAWWBW ยาวกวา? WBWWCW อยู? 7 เซนตเิ มตร และ AWWWCW ยาวกวา? AWWWBW
อยู? 1 เซนตเิ มตร จงหาความยาวของ WAWWBW, WBWWCW และ AWWWCW
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เอกสารประกอบการสอนวชิ าคณิตศาสตรพ2 น้ื ฐาน ม.3 เทอม 1 38
4. รูปสี่เหลยี่ มจัตรุ ัส ABCD และรูปสเ่ี หล่ยี มมุมมาก PQRS มีพ้นื ทเ่ี ทา? กนั และมีขนาด ดงั รูป
จงหาขนาดของรูปส่เี หล่ยี มแตล? ะรูป
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5. ถงั น้ำทรงสี่เหลี่ยมมุมฉากใบหน่ึงมพี ้นื ทกี่ 1นถังเปSน 120 ตารางน้วิ ความยาวรอบปากถังภายในยาว 46 นว้ิ
ถ1าถังใบน้ีจนุ ้ำได1 720 ลูกบาศกน9 ว้ิ จงหาขนาดภายในของถังใบนี้
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เอกสารประกอบการสอนวชิ าคณติ ศาสตรพ2 น้ื ฐาน ม.3 เทอม 1 39
6. กรอบรูปไมส1 กั สำหรบั ใสร? ูปขนาด 24 × 30 ตารางน้ิว มพี ้ืนทโี่ ดยรอบของสว? นทเี่ ปนS ไมส1 ักดา1 นหนา1
ของกรอบรูปเท?ากบั 496 ตารางนิว้ จงหาว?าไม1ทที่ ำกรอบรปู กว1างเทา? ไร
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7. พรกิ หอมต1องการสร1างกรงกระตา? ยใหม1 ีเนื้อท่ี 55 ตารางเมตร ติดกบั รัว้ บ1าน ดงั รปู ถา1 ความยาวของด1าน
ทง้ั สามของกรงกระต?ายรวมกันเปนS 21 เมตร จงหาความกว1างและความยาวของกรงกระตา? ยน้ี
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เอกสารประกอบการสอนวิชาคณิตศาสตรพ2 ้ืนฐาน ม.3 เทอม 1 40
8. กำนันชายปลูกต1นมะยงชิดเรยี งเปSนแถวไว1ในสวน 2,000 ตน1 โดยที่แตล? ะแถวมีจำนวนต1นมะยงชิดเทา? กัน
ถ1าจำนวนต1นมะยงชดิ ในแตล? ะแถวนอ1 ยกวา? จำนวนแถวอย?ู 10 จงหาวา? กำนันชายปลูกต1นมะยงชิดไวก1 ีแ่ ถว
และแถวละก่ตี น1
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เอกสารประกอบการสอนวชิ าคณิตศาสตรพ2 น้ื ฐาน ม.3 เทอม 1 41
แบบฝ<กหัดทา7 ยบท เรอ่ื ง สมการกำลังสองตัวแปรเดียว
1. จงแกส1 มการกำลังสองตวั แปรเดยี วต?อไปนี้
1) x! + 21x = −110 2) −35 = 2y − y!
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3) y(y − 9) = 36 4) 2 + 8x! + 8x = 0
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5) x! + 5x = 0 6) −3y! = 48y + 144
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7) x(x + 5) = 2x 8) 3y! + 19y + 225 = 29 − 9y + 2y!
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9) 3y(5y − 1) = 4(10 − 2y) 10) (2x + 1)! = (2 − x)!
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เอกสารประกอบการสอนวิชาคณิตศาสตรพ2 น้ื ฐาน ม.3 เทอม 1 42
2. โฟกัสมอี ายมุ ากกว?าโฟโต1อยู? 4 ปb ถา1 กำลงั สองของผลรวมของอายุของทง้ั สองคนเทา? กบั 256
อยากทราบวา? โฟโต1มีอายุเทา? ไร
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3. พนื้ ห1องเรยี นหอ1 งหนงึ่ มีพนื้ ท่ี 180 ตารางเมตร และมดี า1 นยาวยาวกว?าดา1 นกวา1 ง 3 เมตร
หอ1 งเรยี นนีก้ วา1 งและยาวก่เี มตร
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4. ปbน้ี คุณพอ? มอี ายุเปSน 3 เทา? ของใยไหม ถา1 สิบปbที่แล1ว กำลงั สองของอายุคณุ พ?อมากกวา? กำลงั สองของ
อายใุ ยไหมอยู? 1,200 อยากทราบว?า ปbนใ้ี ยไหมมอี ายเุ ทา? ไร
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เอกสารประกอบการสอนวชิ าคณติ ศาสตรพ2 ื้นฐาน ม.3 เทอม 1 43
5. ลกู เสือ 2 หมู? เร่มิ ออกเดินทางไกลพร1อมกนั จากโรงเรียนไปยงั ค?ายพกั แรม ตามแผนผังดงั รปู ถา1 ลกู เสอื หมท?ู ี่ 1
ออกเดินไปทางทศิ ใต1 และลูกเสอื หม?ูที่ 2 ออกเดินไปทางทิศตะวันออก เมื่อเวลาผ?านไป 40 นาที ลูกเสอื หม?ทู ่ี 2
เดนิ ทางไดร1 ะยะทางมากกวา? ลูกเสอื หมู?ท่ี 1 อยู? 1 กโิ ลเมตร และลกู เสือทงั้ สองหมอ?ู ย?หู ?างกนั 5 กิโลเมตร
อยากทราบว?าลกู เสอื หมู?ที่ 1 เดินทางมาแล1วกก่ี ิโลเมตร
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6. ลุงปาลม9 ต1องการทำสระว?ายน้ำ เพือ่ ใหเ1 ดก็ ๆ ในหมูบ? 1านมาฝกb ว?ายน้ำ โดยลุงปาลม9 ออกแบบสระว?ายน้ำ
ให1มีสว? นตืน้ และส?วนลึก เปนS 1.20 เมตร และ 2.80 เมตร ตามลำดับ ดังรูป ถ1าดา1 นยาวของสระว?ายนำ้
ยาวน1อยกว?า 2 เท?าของดา1 นกวา1 งอย?ู 5 เมตร และเมอ่ื เตมิ น้ำจนเต็มสระแล1ว จะใช1น้ำ 1,400 ลกู บาศก9เมตร
อยากทราบวา? สระว?ายน้ำนยี้ าวกเี่ มตรและกวา1 งกีเ่ มตร
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เอกสารประกอบการสอนวิชาคณติ ศาสตรพ2 น้ื ฐาน ม.3 เทอม 1 44
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7. โฟลค9 ซองลงแข?งขนั ปนcW จักรยานรายการหน่ึง ซึ่งมรี ะยะทาง 75 กโิ ลเมตร ถ1าในการแข?งขันน้ี
โฟลค9 ซองปWนc จักรยานใหม1 ีอัตราเรว็ เฉลี่ยมากกวา? ปกติ 5 กโิ ลเมตร/ชั่วโมง เขาจะใช1เวลาน1อยลง 30 นาที
อยากทราบว?า โดยปกติแล1วโฟล9คซองปWcนจักรยานดว1 ยอัตราเร็วเฉลย่ี เทา? ไร
ระยะทาง (กม.) อตั ราเรว็ เฉลย่ี เวลาท่ีใช2 (ชม.)
การปนBC จกั รยาน (กม./ชม.)
ของโฟล%คซอง
ปนWc ปกติ
ปWนc แขง? ขัน
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เอกสารประกอบการสอนวิชาคณติ ศาสตรพ2 ื้นฐาน ม.3 เทอม 1 45
8. ภาคณี มีท่ีดนิ รูปส่ีเหล่ียมจตั รุ ัสอยส?ู องแปลงไม?ติดกนั แต?ละแปลงลอ1 มร้วั ดว1 ยลวดหนามสี่ชนั้ ดงั รปู
ถา1 ทีด่ ินทั้งสองแปลงมเี น้ือที่รวมกันเปนS 170 ตารางวา และใชล1 วดหนามท้งั หมด 576 เมตร
อยากทราบวา? ที่ดินของภาคีณแตล? ะแปลงมเี น้ือท่เี ทา? ไร
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เอกสารประกอบการสอนวิชาคณิตศาสตรพ2 น้ื ฐาน ม.3 เทอม 1 46
แบบฝก< หัด 4.1 เรื่อง รปู เรขาคณติ ท่คี ลา7 ยกนั
1. รปู ส่ีเหลยี่ มในแตล? ะขอ1 ตอ? ไปนี้เปSนรูปท่ีคลา1 ยกนั หรือไม? เพราะเหตุใด
1)
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2)
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2. รูปสามเหลี่ยมดา1 นเทา? สองรูปใด ๆ เปSนรปู ท่ีคล1ายกนั หรือไม? เพราะเหตใุ ด
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เอกสารประกอบการสอนวิชาคณติ ศาสตรพ2 ้ืนฐาน ม.3 เทอม 1 47
3. รปู สามเหล่ียมหน1าจั่วสองรูปใด ๆ เปนS รูปทคี่ ลา1 ยกันหรือไม? เพราะเหตใุ ด
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4. รปู สเี่ หลยี่ มจตั ุรสั สองรปู ใด ๆ เปSนรูปทคี่ ลา1 ยกันหรอื ไม? เพราะเหตุใด
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5. รปู สเ่ี หล่ียมขนมเปbยกปูนสองรปู ใด ๆ เปนS รูปทีค่ ล1ายกันหรือไม? เพราะเหตุใด
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6. รูปหกเหลีย่ มดา1 นเท?าสองรปู ใด ๆ เปSนรูปที่คลา1 ยกนั หรือไม? เพราะเหตุใด
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เอกสารประกอบการสอนวิชาคณติ ศาสตรพ2 ้นื ฐาน ม.3 เทอม 1 48
7. รปู หกเหล่ยี มด1านเทา? มุมเท?าสองรปู ใด ๆ เปนS รูปท่คี ลา1 ยกันหรือไม? เพราะเหตุใด
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8. รปู สเี่ หลย่ี มสองรปู ทมี่ ีความยาวของเสน1 รอบรปู เท?ากัน เปนS รูปทีค่ ลา1 ยกันหรอื ไม? เพราะเหตุใด
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9. รปู สามเหล่ียมสองรูปใด ๆ ที่มีพ้ืนท่เี ท?ากัน เปนS รปู ทค่ี ลา1 ยกันหรือไม? เพราะเหตุใด
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10. จากรปู กำหนดให1 RICH ~ BANK จงหาขนาดของมุมทุกมุมที่ไมไ? ดร1 ะบุไว1ในรปู
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เอกสารประกอบการสอนวชิ าคณิตศาสตรพ2 ้ืนฐาน ม.3 เทอม 1 49
11. จากรปู กำหนดให1 COLD ~ WARM จงหา
1) ขนาดของ Ce และ AV
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2) ความยาวของดา1 น LD และความยาวของดา1 น MW
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3) อัตราส?วนของความยาวรอบรปู ของรูปสีเ่ หล่ียมทั้งสองรปู
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เอกสารประกอบการสอนวิชาคณิตศาสตรพ2 ื้นฐาน ม.3 เทอม 1 50
12.
จากรูป รปู สเ่ี หลี่ยมคางหมูทกุ รปู เปนS รปู ทคี่ ล1ายกนั จงหา
1) ขนาดของมุมแต?ละมมุ ของรูปส่เี หล่ยี มคางหมู ABCD
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2) ถา1 WAWWBW ยาว x หนว? ย แลว1 CWWWDW ยาวก่หี น?วย
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คณิตพื้นฐาน ม.3 เทอม 1
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