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Binadamu siku zote wamekuwa wakigundua

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kwamba vitu vingine ni viferu kuliko vingine.

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Kwa mfano, hiki kipande cha mstari kinaonekana

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kirefu kuliko hiki kipande cha mstari.

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Lakini hiyo tu hairidhishi kulinganisha.

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Unatakiwa kuupima.

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Unataka kujua mstari wa pili ni

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mrefu zaidi kuliko wa kwanza.

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Sasa je tunafanyaje?

11
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Vizuri, inabidi tuwe na kipimo cha urefu.

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Kwa hiyo kama tukichukua hiki ndicho kipimo chetu cha urefu,

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tuchukulie hii ni hatua moja, kisha tutaweza kujua

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je hiyo mistari mingine ina hatua ngapi?

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Kwa hiyo mstari wa kwanza unaonekana kama--

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tunaweza kutumia moja kati ya hivyo vipimo kisha tunaweza kurudia tena,

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unaonekana kama una hatua mbili.

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Wakati huu wa tatu unaonekana kama--

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hebu tuhesabu una hatua 1, 2, 3.

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Kwa hiyo huu una hatua tatu.

21
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Na hapa ninatumia hatua.

22
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Ila wakati mwingine tunatumia sentimita

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pale ambapo kipimo huonekana kama hivi.

24
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Kitaonekana tofauti kulingana na skrini yako.

25
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Au tunaweza kuwa na inchi ambayo inaonekana kama hivi.

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Au tunaweza kutumia futi ambayo

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haiwezi kutosha kwenye hii skrini kulingana na ukubwa wa

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inchi au mita niliouchora.

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Kwa hiyo kuna vizio mbalimbali ambavyo

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unaweza kutumia kupima.

31
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Lakini sasa hivi hebu tufikirie vipimo vingine.

32
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Huu mstari ni upande mmoja.

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Huu ni upande mmoja.

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Kwa nini huu ni upande mmoja?

35
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Kwa sababu tunaweza kupima urefu tu.

36
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Hebu sasa hivi tuone pande mbili.

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Tuangalie pande mbili ambapo

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maumbo yanaweza kuwa na urefu na upana au upana na kimo.

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Hebu tuchukulie haya maumbo mawili hapa ambayo yanaonekana kam a hivi.

40
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Tuchukulie hili ni umbo mojawapo.

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Hili ni umbo la kwanza.

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Kumbuka pia, lina upana na kimo.

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Au unaweza kusema lina upana na urefu,

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kulingana na jinsi utakavyoliangalia.

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Hebu tuchukulie hili umbo hapa.

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Hili ni lingine.

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Na hili hapa ni umbo lingine.

48
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Najaribu kuyachora vizuri.

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Kwa hiyo lina pande mbili.

50
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Tunataka kujua je hizi pande mbili

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zinachukua nafasi ya ukubwa gani?

52
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Au kila umbo lina eneo gani?

53
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Tuyalinganishe tena.

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Hili la pili, kama ukiliangalia kama zulia au mstatili,

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mstatili wa pili unachukua nafasi kubwa ya skrini yangu

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kuliko hili la kwanza, lakini nataka kulipima.

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Je, tutawezaje kulipima?

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Inabidi tena tupate vipimo vya mraba kamili.

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badala ya kizio cha urefu, sasa hivi tuna pande mbili.

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Inabidi tuwe na kipimo cha mraba.

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Kwa hiyo tutakuwa na kipimo chetu cha mraba.

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Na kipimo cha mraba tutatafsiri kama mraba,

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ambapo upande na kimo chake

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vyote vinalingana.

65
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Kwa hiyo huu ni upana wake ambao ni hatua moja na kimo chake ni hatua moja.

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Kwa hiyo tutasema una mita 1 za mraba.

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kwa kawaida hii ni hatua moja.

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Kisha unaweka hii 2 hapa, ikimaanisha kipeuo

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Badala ya kuandika hatua, ingeweza pia

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kuwa sentimita.

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Kwa hiyo itakuwa sentimita 1 za mraba.

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Lakini sasa hivi tunaweza kutumia hili umbo kupima haya maeneo.

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Na kama tulivyosema je miraba mingapi yenye urefu huu

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inaweza kutosha kwenye hii mistari,

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tunaweza kujua ni miraba mingapi kama hii inaweza kutosha hapa?

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Kwa hiyo hapa tunaweza kuchukua moja ya mraba kamili

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na kusema unaweza kutosha nafasi hii.

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Tunahitaji mingi zaidi ili kuweza kutosha umbo lote.

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Tutaweka mraba mwingine pale.

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Na mwingine hapa.

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Tutaweka mraba mwingine hapa.

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Vizuri, miraba 4 imeweza kutosha kwenye umbo hili.

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Kwa hiyo tunaweza kusema kwamba hili umbo lina

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eneo la miraba 4 au miraba 4.

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Vipi kuhusu hili hapa?

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inaonekana linaweza kutosha miraba 1, 2, 3, 4, 5, 6, 7, 8,

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9.

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Kwa hiyo hili hapa lingetosha miraba 9.

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hebu tuendelee.

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00:04:07,710 --> 00:04:09,960
Tunaishi kwenye dunia yenye mwonekano wa pande tatu.

91
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Kwa nini tujizuie na kuona pande mbili tu?

92
00:04:12,320 --> 00:04:14,890
Haya hebu sasa tuone hizo pande tatu.

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00:04:14,890 --> 00:04:16,751
Kwa mara nyingine, watu wanaposema pande tatu,

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wanamaanisha mwonekano wa pande zote tatu.

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Wanazungumzia kuhusu mielekeo mitatu tofauti

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unayoweza kupima vitu.

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Hapa kuna urefu tu.

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Hapa kuna urefu na upana au upana na kimo.

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Na hapa kuna upana na kimo na kina.

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Tuchukulie una umbo kama hili

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na tunaishi kwenye dunia

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yenye mwonekano wa pande tatu

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kisha una umbo lingine kama hili,

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inaonekana kama hili umbo la pili linachukua sehemu kubwa zaidi,

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kuliko umbo la kwanza.

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Linaonekana kama ni kubwa zaidi.

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Lakini tunawezaje kulipima?

108
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Kumbuka, ukubwa ni ile nafasi inayotosha umbo zima

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na kuonekana pande tatu.

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Eneo ni nafasi inayochukuliwa na kitu kwenye pande mbili.

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Urefu ni nafasi inayochukuliwa

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upande mmoja.

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Lakini tunapofikiria kuhusu nafasi

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huwa tunafikiria pande tatu.

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Je wewe ungechukua nafasi kiasi gani hapa duniani

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tunamoishi?

117
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Kama tulivyofanya, tunaweza kutafsiri badala ya

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kizio urefu au eneo,

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tunaweza kutafsiri ukubwa au mchemraba kamili.

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Kwa hiyo hebu tufanye wote.

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Hebu tutafsiri mchemraba wetu.

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Na huu hapa ni mchemraba ambao urefu, upana na kimo chake

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vitakuwa na thamani moja.

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Ngoja nijaribu kuchora mchemraba.

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Na zote zitakuwa na urefu wa hatua moja.

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Kwa hiyo itakuwa kimo cha hatua moja, na kina cha hatua moja

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na upana wa hatua moja.

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Kwa hiyo ili kupima ukubwa,

129
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tunaweza kusema je hii michemraba mingapi inaweza kutosha

130
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kwenye haya maumbo?

131
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kama hili hapa

132
00:06:04,391 --> 00:06:06,140
huwezi kuyaona yote.

133
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Ninaweza kulivunjavunja

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ngoja nihesabau kwanza

135
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ili niweze kufanya vizuri.

136
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Ni vigumu kidogo kuyaona yote

137
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kwa sababu kuna baadhi ya michemiraba ambayo iko nyuma yake.

138
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Lakini ukilichukulia kama matabaka mawili,

139
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Kwa hiyo tabaka la kwanza litaonekana kama hivi.

140
00:06:22,980 --> 00:06:25,060
Tabaka la kwanza litaonekana kama hivi.

141
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Hebu fikiria vitu viwili kama hivi vimegandana

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kwa hiyo hiki cha kwanza kitakuwa na michemiraba 1, 2, 3, 4

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Sasa hili litakuwa na matabaka haya mawili

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yaliyogandana.

145
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Kwa hiyo hapa una hatua za ukubwa 8.

146
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Au ungeweza kuwa na hatua za ukubwa 8.

147
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Na hapa je?

148
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Kama tukijaribu kuyaweka hapa yote--

149
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ngoja nichore.

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151
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Litaonekana kama hivi.

152
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153
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Najaribu kuchora kwa haraka.

154
00:06:58,260 --> 00:07:00,600
Kama tukijaribu kuvitenganisha hapa,

155
00:07:00,600 --> 00:07:04,850
tungeweza kuwa na sehemu tatu zilizobandikwa

156
00:07:04,850 --> 00:07:07,840
kila moja litaonekana kama hivi.

157
00:07:07,840 --> 00:07:10,030
Najaribu kuchora vizuri.

158
00:07:10,030 --> 00:07:14,250
Sehemu tatu ambazo zitaonekana

159
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ninachokichora hapa.

160
00:07:19,220 --> 00:07:22,130
Kwa hiyo litaonekana kama hivi.

161
00:07:22,130 --> 00:07:24,270
Kwa hiyo kama ukichukua haya matabaka yaliyogandishwa pamoja

162
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utapata kitu kama hiki hapa.

163
00:07:26,830 --> 00:07:32,050
Na kila tabaka lina michemiraba 1, 2, 3, 4, 5, 6, 7, 8, 9.

164
00:07:32,050 --> 00:07:38,280
9 mara 3, utapata hatua za ukubwa 27 kwenye hili umbo

165
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hapa.

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Hii itatusaidia

167
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kupima vitu

168
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hususani jinsi tunavyopima vitu katika pande tofauti,

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hususani katika mwonekano wa pande tatu tunaita ukubwa.