Te Whakamahi i te Tātai a Heron: Te Wetewete i te Mea Ngaro ki te Tātaitanga o te Horahanga Tapatoru
Ko ngā tapatoru ētahi o ngā āhua māmā rawa engari tino taketake o te āhuahanga, ā, kua roa nei e miharo ana ngā tohunga pāngarau ki ō rātou āhuatanga. Ko tētahi āhuatanga whakamiharo ko te tatau i tō rātou horahanga. Ahakoa he tokomaha e mōhio ana ki te tātai taketake (1/2 teitei turanga), he huarahi huatau, kāore i te tino mōhiotia ko te Tātai a Heron, kāore e hiahiatia te mōhiotanga ki te teitei o te tapatoru. Ka ruku hohonu tēnei tuhinga ki te Tātai a Heron, ka tūhuratia tōna hītori, tōna takenga mai, ōna tono, me tōna hiranga i roto i te ao pāngarau.
Te Hītori o te Tātai a te Mātuku
Ko te Tātai a Heron i tapaina i muri i a Hero o Alexandria, he miihini Kariki, he tohunga pāngarau hoki i noho i te rautau tuatahi AD. Ko Hero, e mōhiotia ana ko Heron, he tohunga nui āna mahi i roto i ngā mara maha tae atu ki te miihini, te hau, me te āhuahanga. Ko āna takoha ki te pāngarau, inā koa te tātai mō te whakatau i te horahanga o te tapatoru mā te whakamahi noa i te roa o ōna taha, e whakanuia tonutia ana tae noa ki tēnei rā.
Te Tātai i Whakakitea
Mā te Tātai a Heron ka taea e tātou te tatau i te horahanga o te tapatoru ina mōhio tātou ki te roa o ngā taha e toru. Ko te tātai e homai ana e:
\[ A = \sqrt{s(sa)(sb)(sc)} \]
ko \( s \) te haurua-porowhita o te tapatoru, ā, ko \( a \), \( b \), me \( c \) ngā roa o ngā taha o te tapatoru. Ka tatauhia te haurua-porowhita \( s \) penei:
\[ s = \frac{a + b + c}{2} \]
Te Whakaputa i te Tātai a Heron
Hei te mārama ki te ataahua o te Tātai a Heron, he mea mārama te titiro ki tōna pūtake. Ko tētahi o ngā tikanga tuku iho ko te whakamahi i ngā tuakiri pākoki me te ture o ngā kosinī. Anei tētahi putanga māmā o ngā mahi e whai wāhi ana:
1. Ture o ngā Kohinīni: Me tīmata tātou me te Ture o ngā Kohinīni e pā ana ki tētahi tapatoru me ngā taha \(a\), \(b\), me \(c\):
\[ c^2 = a^2 + b^2 – 2ab \cdot \cos(C) \]
2. Te horahanga mā te ine-toru: Ka taea hoki te tatau i te horahanga o te tapatoru mā te whakamahi i te tātai e pā ana ki te sine:
\[ A = \frac{1}{2}ab \cdot \sin(C) \]
3. Te Hononga o te Kōsina me te Sina: Mai i ngā tuakiri o runga ake nei me te tuakiri Pythagoras trigonometric, ka taea e tātou te whakaatu i te \(\sin(C)\) e ai ki ngā taha o te tapatoru. Mā tēnei ka uru atu ngā taurangi āwhina e whakangawari ana i ngā whārite ki ngā taha o te haurua-perimita me ngā rerekētanga i waenga i te haurua-perimita me te roa o te taha.
4. Te Whakakotahi me te Whakangawari: Whai muri i ngā mahi tapawhā me te whakakapinga, ka arahina te kīanga whakangawari whakamutunga ki te wāhanga a Heron i raro i te pūtake tapawhā:
\[ A = \sqrt{s(sa)(sb)(sc)} \]
Tauira Tono
Tauira 1: Tātaitanga Māmā
Whakaarohia he tapatoru me ngā taha he 7, 8, me te 9 ngā waeine o te roa:
Tātaihia te haurua-tapahi:
\[ s = \frac{7 + 8 + 9}{2} = 12 \]
Whakamahia te Tātai a Heron:
\[ A = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12 \cdot 5 \cdot 4 \cdot 3} = \sqrt{720} = 26.83 \]
Nō reira, ko te rahi o te rohe he tata ki te 26.83 ngā waeine tapawha.
Tauira 2: Raru Whaihua
Me kī he tapatoru te āhua o tētahi wāhi whenua, ā, he 50m, 60m, me te 70m te rahi o ngā taha. Tātaihia tōna horahanga mā te whakamahi i te Tātai a Heron:
Tātaihia te haurua-tapahi:
\[ s = \frac{50 + 60 + 70}{2} = 90 \]
Whakamahia te Tātai a Heron:
\[ A = \sqrt{90(90-50)(90-60)(90-70)} = \sqrt{90 \cdot 40 \cdot 30 \cdot 20} = \sqrt{2160000} = 1469.69 \]
Nō reira, ko te horahanga o te whenua he tata ki te 1469.69 mita pūrua.
Ngā Painga o te Whakamahi i te Tātai a Heron
1. Kāore he Teitei e Hiahiatia: He rerekē ki te tikanga tuku iho e hiahia ana koe ki te turanga me te teitei, ko te Tātai a Heron e hiahia ana ki ngā roa taha anake.
2. Ngā mahi maha: E pā ana ki ngā momo tapatoru katoa – te tapatoru koi, te tapatoru pūngoi, me te tapatoru tika.
3. Whakangāwari: I ngā tono o te ao tūturu ina uaua te tatau i te teitei, mā te mōhio ki ngā roa taha anake ka tino māmā ake te raruraru.
Whakamahinga Mahi
1. Matawhenua me te Mahere Whenua: Te tatau i te horahanga o ngā poraka whenua tapatoru, o ngā rohe raraunga rangahau āhua koretake rānei.
2. Hangarau: He mea nui mō ngā raruraru hangarau hanganga me te miihini e uru ana ngā wāhanga tapatoru.
3. Whakairoiro Rorohiko: Whakamahia i roto i ngā rauropi mō te whakaputa kupenga, te whakaatu, me te arotau i ngā tātaitanga āhuahanga.
Nga herenga pea
Ahakoa te kaha rawa o te Tātai a Heron, he iti noa iho ōna herenga. Hei tauira:
– Te Pūmautanga Tau: Mō ngā tapatoru he tino nui, he tino iti rānei ngā roa taha, tera pea ka raru te tatau i te tika o ngā ira-mīngi.
– Te Uauatanga mō ngā Tapatoru Kore: Kua hangaia te Tātai a Heron mō ngā tapatoru, ā, kāore e pā tika ki te kore e whakarerekētia ki ētahi atu tapatoru.
Opaniraa
He tohu te Tātai a Heron i te huatau o te pāngarau, e whakaatu ana i te huarahi e taea ai te whakaoti rapanga uaua mā te auahatanga ngāwari. Ahakoa he ākonga koe e tūhura ana i te āhuahanga, he miihini e aro ana ki ngā rapanga o te ao tūturu, he kaitango whenua rānei e mahere ana i ngā whenua, ka whakaratohia e te Tātai a Heron he tikanga pono, whai hua hoki mō te tatau i te horahanga o tētahi tapatoru mā te whakamahi noa i ngā roa taha. Nā ōna pūtake hītori me tōna hiranga mahi ka waiho hei taonga tūturu i roto i te kete taputapu āhuahanga. Mā te mōhio ki te Tātai a Heron, ehara i te mea ka whakaoti noa koe i ngā panga āhuahanga engari ka maioha hoki koe ki te ataahua mutunga kore o te whakaaro pāngarau.
