Ngā Tauira Pātai me te Kōrero mō ngā Porowhita me ngā Aho
Ko te porowhita tētahi o ngā āhua ā-ira taketake e akohia ana i roto i te pāngarau. Ko tētahi ariā nui e pā ana ki ngā porowhita ko te aho, he rārangi e hono ana i ngā pūwāhi e rua i runga i te porowhita me te kore e tika mā waenganui. He mea nui te mārama ki ngā porowhita me ngā aho ki te ako i te āhuahanga. Ka matapakihia e tēnei tuhinga ētahi tauira raruraru me ngā kōrero e pā ana ki ngā porowhita me ngā aho hei whakahōhonu ake i tō tātou mārama ki ēnei ariā.
Te Māramatanga Taketake
Porowhita
Ko te porowhita he kohinga o ngā pūwāhi katoa i roto i te paparangi e rite ana te tawhiti mai i tētahi pūwāhi e kiia nei ko te pokapū. Mena ko te pokapū o te porowhita ko te pūwāhi O, ā, ko te radius o te porowhita ko te r, ka taea te whakaatu i te porowhita mā te whārite (x – O_x)² + (y – O_y)² = r² i te āhua Cartesian.
Aho kopere
Ko te aho i roto i te porowhita he rārangi e hono ana i ngā pūwāhi e rua o te porowhita. Ko te roa o te aho e kore e whakawhirinaki ki te pūtoro o te porowhita anake engari ki te rahi o te koki pokapū e anga atu ana ki a ia.
Ngā Pātai Tauira me te Kōrero
Pātai 1:
Pātai: Homai he porowhita he 10 cm te whānui, me te aho AB he 16 cm te roa. Tāutuhia te tawhiti poto rawa atu i waenganui o te porowhita ki te aho.
Kōrero:
Hei kimi i te tawhiti mai i waenganui o te porowhita ki te aho, ka taea e tātou te whakamahi i te tātai tapatoru matau i hangaia i waenganui i te pūtoro, te tawhiti mai i waenganui ki te aho, me te haurua o te roa o te aho.
Me kī ko te pūwāhi O te pokapū o te porowhita, ā, ko te pūwāhi P te pūwāhi i waenganui o te aho AB e tū poutū ana ki a AB. Nō reira, ko te OP te tawhiti mai i te pokapū o te porowhita O ki te aho AB.
I roto i te tapatoru OAP (tapatoru matau i P), ka taea e tātou te whakamahi i te Ariā Pythagorean:
OP² + AP² = OA²
E mōhio ana mātou:
– OA (te whānui o te porowhita) = 10 cm
– AB (te roa o te aho kopere) = 16 cm, nō reira AP = 16/2 = 8 cm.
Whakakapia ngā uara e mōhiotia ana ki roto i te whārite:
OP² + 8² = 10²
OP² + 64 = 100
OP² = 100 – 64
OP² = 36
OP = √36
OP = 6
Nō reira, ko te tawhiti poto rawa atu i waenganui o te porowhita ki te aho he 6 cm.
Pātai 2:
Pātai: He 8 cm te whānui o te porowhita me te pokapū O. Ko te aho AB he koki pokapū ∠AOB he 120°. Tātaihia te roa o te aho AB.
Kōrero:
Hei tatau i te roa o te aho e hanga ana i te koki pokapū (θ), ka taea e tātou te whakamahi i te tātai:
AB = 2 × r × sin(θ/2)
Kei hea:
– ko r te radius o te porowhita (8 cm i tēnei pātai)
– Ko θ te koki i hangaia e te aho i waenganui o te porowhita (120° i tēnei rapanga)
Whakakapia ngā uara e mōhiotia ana:
AB = 2 × 8 cm × hara(120°/2)
AB = 16 henimita × sin(60°)
AB = 16 henimita × (√3 / 2)
AB = 8√3 henimita
Nō reira, ko te roa o te aho AB he 8√3 cm.
Pātai 3:
Pātai: He porowhita he 13 cm te whānui, ā, e 5 cm te tawhiti o te aho AB mai i te pokapū o te porowhita. Tātaihia te roa o te aho AB.
Kōrero:
I tēnei wā, ka taea e tātou te whakamahi i te tapatoru matau i hangaia hei kimi i te roa o te aho. Me kī ko te pūwāhi O te pokapū o te porowhita, ko te pūwāhi P te pūwāhi i runga i te aho e tata ana ki te pokapū, ā, ko AB te aho.
E mōhio ana mātou:
– OA (te whānui o te porowhita) = 13 cm
– OP (te tawhiti waenga ki te aho kopere) = 5 cm.
I roto i te tapatoru OAP:
OP² + AP² = OA²
Ka kitea e tātou te AP (te haurua o te roa o te aho):
5² + AP² = 13²
25 + AP² = 169
AP² = 169 – 25
AP² = 144
AP = √144
AP = 12
Nō reira, ko te roa o te aho AB = 2 × AP = 2 × 12 = 24 cm.
Pātai 4:
Pātai: Homai he porowhita he 10 cm te whānui. Ko te roa o te aho AB he 12 cm. Tātaihia te rahi o te koki pokapū ∠AOB.
Kōrero:
I tēnei wā, me whakamahi tātou i te whakahurihanga o te mahi pākoki hei whakatau i te koki pokapū ∠AOB. I runga i te tātai mō te roa o te aho e whakauru ana i te koki pokapū θ, ka taea e tātou te tuhi anō i te tātai hei tatau i a θ:
AB = 2 × r × sin(θ/2)
Kei hea:
– ko r te pūtoro o te porowhita (10 cm)
– Ko AB te roa o te aho kopere (12 cm)
Whakatūria he whārite hei wehe i te sin(θ/2):
12 = 2 × 10 × sin(θ/2)
12 = 20 × sin(θ/2)
hara(θ/2) = 12/20
hara(θ/2) = 0.6
Kimihia te uara o θ/2:
θ/2 = sin^(-1)(0.6)
θ/2 = 36.87°
Nā reira,:
θ = 2 × 36.87° = 73.74°
Nō reira, ko te koki pokapū ∠AOB he 73.74°.
Whakamutunga
Ko te ako i ngā porowhita me ngā aho e hiahia ana kia mārama ki te āhuahanga taketake me te ine whārite. I roto i ngā tauira i runga ake nei, kua kite tātou me pēhea te whakamahi i ngā momo tātai me ngā ariā, pērā i te ariā Pythagorean, ngā mahi ine whārite, me ngā āhuatanga taketake o ngā porowhita, hei whakaoti rapanga.
Mā roto i ēnei mahi whakaharatau, e tūmanakohia ana ka taea te whakatutuki i tētahi māramatanga hohonu ake mō te whanaungatanga i waenga i te pūtoro, te roa o te aho, me te koki pokapū ka puta mai. He mea whai hua te mārama ki tēnei ariā ehara i te mea i roto i te pāngarau kura anake engari i roto hoki i ngā tono o ia rā puta noa i ngā momo mara o te pūtaiao me te hangarau.
