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8-ð àíãè 16 öàã
Ìàòåìàòèêèéí áàãø Ä. Ýíõòóÿà
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Çîðèëãî
Õýðýãöýý, øààðäëàãàÍýãæ õè÷ýýëèéí
õºòºëáºð
Ñýäâèéí òºëºâëºëò
Àãóóëãà
Çîðèëò
Õýðýãëýãäýõ¿¿í
¯íýëýõ æóðàì
Æèøèã äààëãàâàð
¯íýëãýýíèé øàëãóóð
Ä¿ãíýëò
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Õýðýãöýý, øààðäëàãà Ãåîìåòðèéí êóðñûã ã¿íçãèéð¿¿-ëýí ñóäàëæ, öààøèä ñóäëàõ àãóóëãûí áýëòãýë áîëãîõ, îð÷èí òîéðíû þìñ ¿çýãäëèéí çàãâàð áîëãîí àøèãëàæ, òóëãàðñàí àñóóäëûã øèéäâýðëýõ
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Çîðèëãî
ªìíº ¿çñýí ãóðâàëæíû ìýäëýãýý ñýðãýýí áàòàòãàæ, ã¿äãýð äºðâºí ºíöºãò ïàðàëëåëîãðàìì, òðàïåö, òýäãýýðèéí òàëáàéí ìýäëýãèéã øèíýýð á¿òýýæ ýçýìøèõ àðãàä ñóðàëöóóëàõ
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Çîðèëò
• Ãóðâàëæíû ¿íäñýí õýð÷ì¿¿ä áà õýìæèãäýõ¿¿í, òàë, ºíöãèéí õàìààðàë, òàëáàéí ìýäëýãýý ñýðãýýõ äàäëàãà àæèë õèéõ• ĺðâºí ºíöºãòèéã àæèãëàæ, ýñðýã îðøèõ õîñ òàëóóäûí õàðèëöàí áàéðøëûí áîëîìæîîð àíãèëàí õ¿ñíýãòýëæ, çóðæ, òýìäýãëýæ ñóðàõ• Ïàðàëåëîãðàìì, ðîìáî, òýãø ºíöºãò, êâàäðàòûí óõàãäàõóóí ýçýìøèí òýäãýýð ä¿ðñ¿¿ä áîëîí ýëåìåíò¿¿äèéã íü çóðæ òýìäýãëýõ, ìºí ÷àíàðóóäûã íü íýýí îëæ, áè÷èæ òýìäýãëýæ ñóðàõ• ÿäãýð äºðâºí ºíöºãò¿¿äèéí òàëáàé îëîõ àðãóóäûã ¿ãýýð áîëîí òîìú¸îãîîð èëýðõèéëýõ, õýðýãëýõ ÷àäâàð ýçýìøèõ• Äýýðõ çîðèëòóóäûã õýðýãæ¿¿ëýõýä äàñãàë, äàäëàãà àæëûã áèå äààí õèéæ ÷àääàã áîëîõ
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Àãóóëãàªìíº ñóäàëñàí Îäîî ñóäëàõ
Öààøèä ñóäëàõ
Ãóðâàëæèí, ãóðâàëæíû ¿íäñýí äºðâºí õýð÷èì
Ãóðâàëæíû ¿íäñýí õýìæèã- äýõ¿¿í
Òýãø ºíöºãò ãóðâàëæíû òàë áà ºíöãèéí õàìààðàë
• ÿäãýð äºðâºí ºí- öºãò,ò¿¿íèé ýëåìåíò
• ÿäãýð äºðâºí ºíöºãòèéí àíãèëàë
• Ïàðàëëåëîãðàìì, ðîìáî, òýãø ºíöºãò, êâàäðàò
• Òðàïåö, àäèë õàæóóò áîëîí òýãø ºíöºãò
• Äóðûí ã¿äãýð äºðâºí ºíöºãò, ïàðàëëåëîãðàìì, ðîìáî, òðàïåöûí òàëáàé
Òîéðîãò áàãòñàí áà áàãòààñàí ã¿äãýð äºðâºí ºíöºãò
Îëîí ºíöºãò, çºâ îëîí ºíöºãò
Îëîí ºíöºãòèéí òàëáàé
Òîéðîã, äóãóé, äóãóéí òàëáàé
Âåêòîð, ò¿¿íèé ¿éëäë¿¿ä
Âåêòîð àðãûí õýðýãëýý
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“ĺðâºí ºíöºãò” ñýäâèéã ñóäëàõ õè÷ýýëèéí òºëºâëºëò
¹ ÑýäýâÖàã
Àãóóëãà
Ñóäëàãäàõóóí Óð óõààí
1 Ãóðâàëæèí 1
Ãóðâàëæèí, ãóðâàëæíû òàë,ºíöãèéí õàìààð-ëûí ìýäëýã ñýðãýýõ
Ãóðâàëæíû ìýäëýãèéã õýðýãëýí àñóóäàë
øèéä-âýðëýõ äàäëàãà ýçýìøèõ
2 Ôàëåñûí òåîðåì 1Ôàëåñûí òåîðåìûí
áàòàëãàà , õýðýãëýý
Ôàëåñûí òåîðåìûí áàòàëãààã õèéõ,
àñóóäàë øèéäâýðëýõýä
àøèãëàõ
3
ÿäãýð äºðâºí ºíöºãò,ã¿äãýð äºðâºí ºíöºã-òèéí àíãèëàë
1
ÿäãýð 4 ºíöºãò áà ò¿¿íèé ýëåìåíòèéí
óõàãäàõóóíÿäãýð äºðâºí ºíöºã-òèéí ¿íäñýí 3 òºðºë
áîëîõ ïàðàëëåëî-ãðàìì,òðàïåö, àëèâàà
4 ºíöºãòèéã òîäîðõîéëîõ øèíæ¿¿ä
ÿäãýð 4 ºíöºãò áà ò¿¿-íèé ýëåìåíòèéã òîäîð-õîéëîõ, çóðàõ, òýìäýã-ëýõ, õýìæèõ,
áîäîõ àðãà ýçýìøèõ,äóðûí äºðâºí
ºíöºãò, ïàðàëëåëîãðàìì, òðàïåöèéã òºðëèéí øèí-
æýýð ÿëãàí õ¿ñíýãòýëæ, çóðàõ, òýìäýãëýõ, áè÷èæ
òîäîðõîéëîõ
2ÌÀ3\1à),3ç),è)
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4Ïàðàëëåëî-
ãðàìì2
Ïàðàëëåëîãðàìì áà ò¿¿íèé ýëåìåíò¿¿-äèéí òîäîðõîéëîãäîõ øèíæ,
¿íäñýí ÷àíàðóóä
Ïàðàëëåëîãðàìì áà ò¿¿íèé ýëåìåíò¿¿äèéã
òîäîðõîéëîõ, çóðàõ, íýðëýõ, òýìäýãëýõ, ¿íäñýí ÷àíàðóóäûã íýýí èëð¿¿ëæ ¿ãýýð áîëîí òýìäýãëýëýýð
èëýðõèéëýí òîìú¸îëîõ, òýäãýýðèéã
õýðýãëýæ áîäëîãî áîäîõ áà àñóóäëûã øèéäýõ àðãóóäûã
ýçýìø¿¿ëýõ
5Ðîìáî,
òýãø ºíöºãò, êâàäðàò
3
Ðîìáî, òýãø ºíöºãò, êâàäðàò áà
òýäãýýðèéí ¿íäñýí ÷àíàðóóä
Ýäãýýð ä¿ðñ¿¿ä áîëîí òýäãýýðèéí ýëåìåíò¿¿-äèéã íü ä¿ðñëýõ, òýì-
äýãëýëýýð èëýðõèéëýí òîìú¸îëîõ, òýäãýýðèéã õýðýãëýí
áîäëîãî äàñãàë àæèëëàæ, àñóóäëûã
øèéäâýðëýõ
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6 Òðàïåö 2
Òðàïåö, àäèë õàæóóò òðàïåö, òýãø ºíöºãò òðàïåö áîëîí òýäãýý-
ðèéí ýëåìåíòèéí òîäîðõîéëîõ øèíæ¿¿ä
Òðàïåö áà ò¿¿íèé ýëå-ìåíò¿¿äèéã òîäîðõîéëîõ,
çóðàõ, íýðëýõ, òýìäýãëýõ, ¿íäñýí ÷àíàðóóäûã íýýí
èëð¿¿ëæ, òîìú¸îëîõ, òýä-ãýýðèéã õýðýãëýæ
áîäëîãî áîäîõ áà àñóóäëûã øèéäýõ àðãóóäûã
ýçýìø¿¿ëýõ
7 Òåñò 2ĺðâºí ºíöºãò ñýäâèéí
àãóóëãà
Ýíý ñýäâýýð ýçýìøâýë çîõèõ ìýäëýã, ÷àäâàðûí
öîãö
8ÿäãýð äºðâºí
ºíöºãòèéí òàëáàé
3
ÿäãýð äºðâºí ºíöºãò áîëîí ïàðàëëåëî-ãðàììûí
òàëáàé îëîõ àðãóóä
Ýäãýýð àðãóóäûã àæèãëàí íýýæ îëîõ, ¿ãýýð
áîëîí òîìú¸îãîîð èëýðõèéëýõ, òýäãýýðèéã
õýðýãëýí áîäëîãî áîäîõ, àñóóäëûã øèéäâýðëýõ
9 Õÿíàëò ¿íýëãýý 1ĺðâºí ºíöºãò ñýäâèéí
àãóóëãà
Ýíý ñýäâýýð ýçýìøâýë çîõèõ ìýäëýã, ÷àäâàðûí
öîãö
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Ôèçè
ê
2Ô2\2ã,ä7æ,ì
Ìýäýýëýë ç¿é 2ÌÇ3\1
Îþóíû òîãëîîì
Ä¿ðñëý
õ
óðëà
ã
Ìàòåìàòèê
2Ô2/3
2ÄÓ3\6
2
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Ëèíç
Öóãëóóëàã÷ Ñàðíèóëàã÷
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ÆèãäÆèãä õóðäñàõ Æèãä óäààøðàõ
v,m/c
10
0 2 t,c
v,m/c
10
5
0 20 t,c
v,m/c
10
5
0 20 t,c
00 v 0tv
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tvl 2
2
0
attvl
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lS
v,m/c
10
0 5 10 15 20 t,c
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Ïàðàëëåëîãðàììûí ä¿ðýì
F1+F2
F1
F2
10
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Òýìäýãëýãýý
Ýõëýë /òºãñãºë/
Îðîëò, ãàðàëò
Øóóä áèåëýõ
¿éëäýë ͺõöºë øàëãàõ
Òóñëàõ àëãîðèòì
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Ýõëýë
n,mN
S=n+m
Òºãñãºë
S
Æèøýý1
Õî¸ð íàòóðàë òîîíû íèéëáýð îëîõ àëãîðèòì
Ýõëýë
n=4,m=6
S=4+6
Òºãñãºë
S
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Á¿ëýã ¿éëäýë 1 Á¿ëýã ¿éëäýë 2
ͺõöºë
10
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Àìüäðàë àõóéä òîõèîëäîõ 4 ºíöºãò¿¿ä
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Ç¿éõ óðëàã áà äºðâºí ºíöºãò
10
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Êîä Øàëãóóð
2ÌÀ3:Ê1\1Îð÷èí òîéðíûõîî þìñûã àæèãëààä äºðâºí ºíöºãò (òýãø
ºíöºãò, ïàðàëëåëîãðàìì, ðîìáî, êâàäðàò, òðàïåö ã.ì) îëîõ, õýìæýýã íü í¿äýí áàðèìæààãààð ãàðãàæ ÿðèõ
2ÌÀ3:Ê1\2 Àíãèéí òààç, øàë, õàíà, öîíõ, õààëãà áóäàõ òîîöîîíû çóðàã òºñºë ãàðãàí òàíèëöóóëàõ
2ÌÀ3:Ê2\1-3Ïàðàëëåëîãðàìì, òýãø ºíöºãò, ðîìáî, êâàäðàò, òðàïåöûã
áîäèò þìñûí çàãâàð áîëãîí ÷àíàðóóäûã íü àøèãëàí áîäëîãî áîäîõ
2ÌÀ3:Ê3\1 ĺðâºí ºíöºãòèéí ÷àíàðóóäûã çóðàã, òýìäýãëýë, òîìú¸îãîîð èëýðõèéëýí ÿðèõ
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2ÌÀ3:Ê3\2-3Ä¿ðñèéí ýëåìåíò õîîðîíäûí õàìààðëûí ÷àíàðóóäûã
òîìú¸îëîí áàòëàõ, áàòàëãààã çóðàã, òîâ÷ áè÷ëýãýýð õèéõ, òýäãýýðèéã àøèãëàí áîäëîãî çîõèîõ
2ÌÀ3:Ê3\4Áàéãóóëàõ áîäëîãî áîäîõ, àíàëèç õèéõ, àñóóäëûã øèéäâýðëýõýä äºðâºí ºíöºãò¿¿äèéí ýëåìåíò¿¿äèéí
÷àíàðûã àøèãëàõ
2ÌÀ3:Ê4\1ĺðâºí ºíöºãò ò¿¿íèé ýëåìåíò õîîðîíäèéí ø¿òýëöýë,
÷àíàðààñ ¿ã, çóðàã, òýìäýãëýë, òîìú¸îãîîð èëýðõèéëýõ, ¿íýí çºâ áîëîõûã ¿íäýñëýõ
2ÌÀ3:Ê4\2Òåîðåì, ÷àíàðûã áàòëàõ, òîìú¸îíû ãàðãàëãàà õèéõ
2
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¹ ÍýðñÕÿíàëòûí
àæèëÒåñ
òªºðèéí
¿íýëãýý
Åðºíõèé
¿íýëãýý
1
2
Ä¿í
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Ïàðàëëåëîãðàììûí ºãºãäñºí õýëáýð¿¿äýýñ
àëü ÷àäàìæèéã ýçýìøñýíýý òýìäýãëýíý ¿¿
Ïàðàëëåëîãðàììûí õýëáýð¿¿ä
ªºðèéí ¿íýëãýý,%
Ïàðàëëåëî-ãðàìì
Òýãø ºíöºãò Ðîìáî Êâàäðàò
Çóðæ, ä¿ðñëýõ 100 Ýëåìåíò¿¿äèéã íýðëýõ ×àíàðóóäûã òîìú¸îëîí õýëýõ
Òîìú¸îãîîð áè÷èõ 90 Õÿëáàð òîõèîëäîëä àøèãëàõ
Òàë áà ºíöãèéí õàìààðëûã õýðýãëýí àñóóäàë øèéäýõ
Áîäèò þìñûí çàãâàð áîëãîí àøèãëàõ
Õ¿íäýâòýð áîäëîãî áîäîõ
2
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Ïàðàëëåëîãðàììûí ºãºãäñºí õýëáýð¿¿ä íü äîîðõ ÷àíàðûã õàíãàõ
ýñýõèéã õàðèóëæ áè÷ýýðýé.
Ïàðàëëåëîãðàììûí õýëáýð¿¿äÏàðàëëåëî-ãðàìì
Òýãø ºíöºãò Ðîìáî Êâàäðàò
Ýñðýã òàëóóä ïàðàëëåëü òèéì
Äèàãîíàëü íü ò¿¿íèéã 2 òýíö¿¿ ãóðâàëæèíä õóâààíà
Ýñðýã òàëóóä òýíö¿¿
Á¿õ òàë íü òýíö¿¿
Ýñðýã ºíöã¿¿ä òýíö¿¿ ¿ã¿é
Á¿õ äîòîîä ºíöã¿¿ä òýíö¿¿
Á¿õ äîòîîä ºíöã¿¿ä òýãø Íýã òàëä íàëñàí ºíöã¿¿äèéí íèéëáýð 180ãðàäóñ
Äèàãîíàëèóä íü îãòëîëö-ëûíõîî öýã¿¿äýýð õàãàñëàí õóâààãäàíà
Äèàãîíàëèóä íü òýíö¿¿
Äèàãîíàëèóä íü õàðèëöàí ïåðïåíäèêóëÿð
Äèàãîíàëèóä íü ºíöã¿¿äèéã íü õàãàñëàí õóâààíà
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ĺðâºí ºíöºãò ò¿¿íèé ýëåìåíò
Íýð Áèåò æèøýý,
çóðàã
Ãåîìåòð ä¿ðñëýë,
òýìäýãëýæ íýðëýõ
Òîäîðõîéëîëò
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Õýðýãëýãäýõ¿¿í
Öàõèì ¿ç¿¿ëýí
Òåñò
Äàäëàãà àæèë
Òåñò
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Íýð ÒîäîðõîéëîëòÇóðàã,
ä¿ðñëýë
×àíàð
¯ãýýð Òîìú¸îãîîð
Òýãøºíöºãò
Ðîìáî
Êâàäðàò
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Òýãø ºíöºãò Ðîìáî Êâàäðàò
Òýãø ºíöºãòèéí
ïåðèìåòð 36ñì áà íýã òàë íü 8ñì áîë íºãºº
òàëûã îëîîðîé.
Ðîìáûí òàë 13ñì áà íýã äèàãîíàëü íü
10ñì áîë ò¿¿íèé íºãºº äèàãîíàëèéã
îëîîðîé.
Êâàäðàòûí ïåðèìåòð 40ñì
áîë ò¿¿íèé òàëáàéã îëîîðîé.
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2
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Äàðààõ ä¿ðñ¿¿äèéã ýâë¿¿ëæ á¿òýýãýýðýé
10
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40
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40
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ĺðâºí ºíöºãò ñýäâèéã ñóäàëñíààð ñóðàëöàã÷èä ãåîìåòðèéí ñýäâèéã öààøèä ñóäëàõ îíîë, àðãà ç¿éí áýëòãýëèéã õàíãàæ, àìüäðàë àõóéí çàðèì àñóóäëûã ìàòåìàòèê çàãâàðò îðóóëàí øèéäýõ ÷àäâàðòàé áîëíî. ̺í ôèçèê, ä¿ðñëýõ óðëàã ãýõ çýðýã õè÷ýýëýýð ñóäëàõ çàðèì ¿çýãäýë, þìñûí ìºí ÷àíàðûã èë¿¿ ã¿íçãèé îéëãîõ áîëíî. Èíãýñíýýð àñóóäàëä îëîí òàëààñ íü õàíäàõ îëîí óðãàëü÷ ¿çëèéã òºëºâø¿¿ëýõ þì.
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Ýíý õºòºëáºðèéã áîëîâñðóóëàõàä àðãà ç¿éí çºâëºãºº ºãñºí ìàòåìàòèêèéí áàãø
Ö. Ýðäýíýñóâä, Á. Ãàíáàÿð, Æ. Äîëãîðìàà, Ä. Ýðäýíý÷èìýã; ôèçèêèéí áàãø Ö.
Ãàí÷èìýã, Á. Ñàíäàãäîðæ, çóðãèéí áàãø Ë. Öýöãýý áîëîí ìýäýýëýë ç¿éí áàãø Í.
Äýëãýðìàà, À. Ýðäýíý÷èìýã íàðò òàëàðõñàíàà èëýðõèéëüå.