Ngā Whārite Raina Tika i roto i te Āhuahanga
I roto i te āhuahanga me te pāngarau whānui, ko te rārangi tika tētahi o ngā mea tino taketake engari he mea tino nui. Tata ki ngā ariā āhuahanga katoa—mai i ngā koki me ngā āhua papatahi ki ngā panonitanga—e pā ana ki ngā rārangi. Nō reira, mā te mārama ki te whārite o te rārangi tika ka whakaratohia he tūāpapa pakari mō te ako i ngā kaupapa matatau ake, pērā i ngā pūnaha whārite rārangi, te āhuahanga tātari, te tātaitai, me te ahupūngao. Ka matapakihia e tēnei tuhinga te whakamāramatanga, ngā āhua o te whārite rārangi tika, me pēhea te whakatau, me ngā tauira o tōna whakamahinga i roto i te āhuahanga.
1. Te Mārama ki te Whārite o te Raina Tika
I roto i ngā kupu māmā noa iho, ko te whārite o te rārangi tika he whanaungatanga pāngarau e whakaahua ana i ngā pūwāhi katoa e takoto ana i runga i te rārangi i roto i te papa taunga. I roto i te pūnaha taunga Cartesian, ka tohuhia ia pūwāhi hei takirua raupapa \((x, y)\). Mena ka tutuki i tētahi pūwāhi tētahi whārite, ka takoto ia i runga i te rārangi e tohuhia ana e taua whārite.
Hei tauira, ko te whārite \(y = 2x + 1\) e tohu ana i te huinga o ngā pūwāhi katoa, ina tāpirihia he uara o \(x\), ka whiwhihia te uara o \(y\) e ai ki te ture. Mēnā ka tuhia e tātou ngā pūwāhi katoa e tutuki ana i tēnei whanaungatanga, ka hanga he rārangi tika.
2. Te Pikinga (Te Pikinga) o te Raina
Ko tētahi ariā nui i roto i te whārite o tētahi rārangi tika ko te pikinga, te pari rānei, e tohuhia ana e \(m\). Mā te pikinga ka whakaatu koe i te pikinga, te hekenga rānei o te rārangi i a ia e neke ana mai i te maui ki te matau.
Ko te whakarōpūtanga e tautuhia ana ko:
\[
m = \frac{\Delta y}{\Delta x} = \frac{y_2 – y_1}{x_2 – x_1}
\]
ko \((x_1, y_1)\) me \((x_2, y_2)\) he pūwāhi motuhake e rua i te rārangi.
Whakamāramatanga o te rōnaki:
– Ki te mea ko te \(m > 0\), ka piki te rārangi mai i te maui ki te matau.
– Ki te \(m < 0\), ka heke te rārangi mai i te maui ki te matau. - Ki te \(m = 0\), he whakapae te rārangi. - Ki te poutū te rārangi, kāore i te tautuhia te pikinga nā te mea \(\Delta x = 0\). He mahi ā-ira anō hoki tā ngā pikinga: e rua ngā rārangi whakarara he pikinga ōrite, ko ngā rārangi poutū ia e rua he whanaungatanga pikinga \(m_1 \cdot m_2 = -1\) (mēnā kāore i te poutū/whakapae, me whakahaere motuhake). 3. Ngā Āhua o ngā Whārite Raina He maha ngā āhua o ngā whārite raina e whakamahia pinepinetia ana, i runga i ngā mōhiohio e wātea ana. a) Āhua Piki-Whakatau Ko te āhua tino noa ko: \[ y = mx + c \] kei hea: - \(m\) = piki o te rārangi - \(c\) = piki-whakatau y (te uara o \(y\) ina \(x = 0\)) Tauira: \(y = 3x - 2\) ko te tikanga ko te piki he 3, ā, ka whakawhiti i te tuaka \(y\) i \(-2\). b) Āhua Whānui Ko te āhua whānui o te whārite o te rārangi ko: \[ Ax + By + C = 0 \] kei hea ko \(A, B, C\) he tau tūturu, ā, ehara a \(A\) me \(B\) i te kore. He maha ngā wā ka whakamahia tēnei āhua mō te tātari āhuahanga, hei tauira te whakatau i te tawhiti mai i tētahi pūwāhi ki tētahi rārangi, te kimi rānei i te pūwāhi e whakawhiti ana i ngā rārangi e rua. Tauira: \(2x + y - 5 = 0\). c) Āhua Piki-Whakatau
Mena e mōhio ana tātou ki tētahi pūwāhi \((x_1, y_1)\) me tētahi pari \(m\), ko te āhua koia tēnei: \[ y - y_1 = m(x - x_1) \] He tino whai hua tēnei āhua ina he raraunga kei a tātou hei pūwāhi i runga i tētahi rārangi me tōna pari. Hei tauira: he rārangi mā \((2, 3)\) me te pari o te 4: \[ y - 3 = 4(x - 2) \] ka taea te whakangawari ki \(y = 4x - 5\). d) Āhua Pūwāhi-Rua Mena e mōhiotia ana ngā pūwāhi e rua \((x_1, y_1)\) me \((x_2, y_2)\), ka taea te tiki i te whārite o te rārangi mai i: \[ \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} \] Ka hono tika tēnei āhua i ngā pūwāhi katoa \((x, y)\) e rārangi ana ki ngā pūwāhi e rua. e) Āhua Taupoki Mena ka whakawhiti te rārangi i te tuaka \(x\) i \((a, 0)\) me te tuaka \(y\)- i \((0, b)\), ko te whārite koia tēnei: \[ \frac{x}{a} + \frac{y}{b} = 1 \] Ka āwhina tēnei āhua i te tirohanga nā te mea ka whakanui i ngā pūwāhi e whakawhiti ana ki ngā tuaka. 4. Te Whakatau i te Whārite o te Raina Tika I roto i te āhuahanga tātari, ka puta te pātai "te whakatau i te whārite o te raina" i runga i ētahi mōhiohio. Anei ētahi āhuatanga noa: a) Ki te mea ko te Pikinga me te Āraitanga \(y\) Mena e mōhiotia ana te pikinga \(m\) me te āraitanga \(c\), whakamahia tika \(y = mx + c\). Tauira: rōnaki \(-2\), āraitanga \(y\) = 3: \[ y = -2x + 3 \] b) Ki te mea e rua ngā pūwāhi i hoatu Hei tauira, ki te mea ko \((1, 2)\) me \((3, 6)\). Pikinga: \[ m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2 \] Whakamahia te pūwāhi \((1, 2)\):
\[ y - 2 = 2(x - 1) \Rightarrow y = 2x \] c) Ngā Rārangi Whakarara, Ā-Porotū rānei - Ngā rārangi whakarara: he pari kotahi. - Ngā rārangi poutū: he pari whakamuri kino (mēnā e tika ana), arā, \(m_2 = -\frac{1}{m_1}\). Tauira: ko te pari o te rārangi \(y = 3x + 1\) he 3. Ko te pari o te rārangi e poutū ana ki a ia he \(-\frac{1}{3}\). Mēnā ka haere te rārangi poutū mā \((0, 2)\): \[ y - 2 = -\frac{1}{3}(x - 0) \Rightarrow y = -\frac{1}{3}x + 2 \] 5. Ngā Whakamahinga i roto i te Āhuahanga Ehara i te mea he mea nui anake te whārite o te rārangi tika i roto i te ā-ira, engari he tino whai hua hoki i roto i te āhuahanga: 1. Te whakatau i te pūwāhi e whiti ai ngā rārangi e rua. Ka whiwhihia te pūwāhi e whiti ai mā te whakaoti i tētahi pūnaha whārite. 2. Te tatau i te tawhiti mai i tētahi pūwāhi ki tētahi rārangi. Mā te āhua whānui \(Ax + By + C = 0\), ko te tawhiti mai i te pūwāhi \((x_0, y_0)\) ki te rārangi ko: \[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \] 3. Te tātari i ngā āhua papatahi. Ka taea te whakaatu i ngā taha o tētahi tapatoru, tapawhā, āhua rānei hei rārangi, kia taea ai te ako i ngā āhuatanga o te āhua mā te whārite o te rārangi. 4. Te whakatau i te wehenga me te teitei o tētahi tapatoru. He poutū te teitei ki tētahi taha kua hoatu, ko te wehenga koki he ture motuhake, ā, ka taea te tatau i ēnei katoa mā te whakamahi i te pari. 6. Whakamutunga Ko te whārite o tētahi rārangi tika he taputapu matua i roto i te āhuahanga tātari hei whakaatu i ngā rārangi i runga i te papa taunga. Mā te mārama ki te pari me ngā momo āhua o te whārite—pērā i te \(y = mx + c\), \(Ax + By + C = 0\), te āhua pari-ira, te āhua pū-rua, me te āhua tuaka-tauwhitinga—ka taea e tātou te tautuhi me te tātari i ngā rārangi. He tino whai hua tēnei pūkenga mō te whakaoti rapanga āhuahanga pērā i te whakatau i ngā pūwāhi whakawhiti, te tatau i ngā tawhiti, me te tirotiro mēnā he whakarara, he poutū rānei ngā rārangi. Hei whakamutunga, ka whakaratohia e tēnei ariā māmā he piriti nui i waenga i te āhuahanga tirohanga me te arorangi pūnaha.
