Litekanyo tsa Mola o Otlolohileng ho Geometry
Ho jiometri le lipalo ka kakaretso, mola o otlolohileng ke e 'ngoe ea lintho tsa motheo empa e le tsa bohlokoa ka ho fetisisa. Hoo e ka bang likhopolo tsohle tsa jiometri—ho tloha likhutlong le lipalo tse bataletseng ho ea liphetohong—li amana le mela. Ka hona, ho utloisisa equation ea mola o otlolohileng ho fana ka motheo o tiileng oa ho ithuta lihlooho tse tsoetseng pele haholoanyane, joalo ka litsamaiso tsa li-equation tse otlolohileng, jiometri ea tlhahlobo, calculus le fisiks. Sengoloa sena se tšohla tlhaloso, mefuta ea equation ea mola o otlolohileng, mokhoa oa ho e fumana, le mehlala ea ts'ebeliso ea eona ho jiometri.
1. Ho Utloisisa Tekanyo ea Mola o Otlolohileng
Ka mantsoe a bonolo, equation ea mola o otlolohileng ke kamano ea lipalo e hlalosang lintlha tsohle tse moleng o lebileng sefofaneng sa coordinate. Sistemeng ea coordinate ea Cartesian, ntlha ka 'ngoe e emeloa e le para e laetsoeng \((x, y)\). Haeba ntlha e khotsofatsa equation e itseng, joale e lutse moleng o emetsoeng ke equation eo.
Mohlala, equation \(y = 2x + 1\) e emela sete ya dintlha tsohle tseo, ha boleng ba \(x\) bo kenngwa, boleng ba \(y\) bo fumanwang ho ya ka molao. Haeba re rala dintlha tsohle tse kgotsofatsang kamano ena, di tla bopa mola o otlolohileng.
2. Gradient (Moedi) oa Mohala
Khopolo ea bohlokoa ho equation ea mola o otlolohileng ke gradient kapa leralla, hangata le bontšoang ke \(m\). Gradient e u bolella hore na mola o nyoloha kapa o oa ka lebelo hakae ha o ntse o tsamaea ho tloha ka letsohong le letšehali ho ea ho le letona.
Gradient e hlalosoa e le:
\[
m = \frac{\Delta y}{\Delta x} = \frac{y_2 – y_1}{x_2 – x_1}
\]
moo \((x_1, y_1)\) le \((x_2, y_2)\) e leng lintlha tse peli tse fapaneng moleng.
Tlhaloso ea gradient:
– Haeba \(m > 0\), mola o nyoloha ho tloha ka letsohong le letshehadi ho ya ho le letona.
– Haeba \(m < 0\), mola o theohela tlase ho tloha ka letsohong le letshehadi ho ya ho le letona. - Haeba \(m = 0\), mola o rapaletse. - Haeba mola o otlolohile, gradient ha e hlaloswe hobane \(\Delta x = 0\). Di-gradients le tsona di na le karolo ya jeometri: mela e mmedi e bapileng e na le gradient e tshwanang, ha mela e mmedi e otlolohileng e na le kamano ya gradient \(m_1 \cdot m_2 = -1\) (ha feela e se e otlolohileng/e otlolohileng, e hlokang ho tshwarwa ka tsela e ikgethang). 3. Mefuta ya Di-Equation tse Mola Ho na le mefuta e mmalwa ya di-equation tse mola tse atisang ho sebediswa, ho itshetlehile ka tlhahisoleseding e fumanehang. a) Sebopeho sa ho Thibela Mothapo Sebopeho se tlwaelehileng haholo ke: \[ y = mx + c \] moo: - \(m\) = leralla la mola - \(c\) = y-thibela (boleng ba \(y\) ha \(x = 0\)) Mohlala: \(y = 3x - 2\) e bolela hore leralla ke 3 mme le kopana le mothapo wa \(y\) ho \(-2\). b) Sebopeho se Akaretsang Sebopeho se akaretsang sa equation ya mola ke: \[ Ax + By + C = 0 \] moo \(A, B, C\) e leng dinomoro tsa nnete mme \(A\) le \(B\) ka bobedi e seng lefela. Sebopeho sena hangata se sebediswa bakeng sa tlhahlobo ya jeometri, mohlala, ho fumana sebaka ho tloha ntlheng ho ya moleng kapa ho fumana ntlha ya kgokelo ya mela e mmedi. Mohlala: \(2x + y - 5 = 0\). c) Sebopeho sa Ntlha-Thibela
Haeba re tseba ntlha \((x_1, y_1)\) le leralla \(m\), sebopeho ke: \[ y - y_1 = m(x - x_1) \] Sebopeho sena se thusa haholo ha re na le data ka sebopeho sa ntlha moleng le leralla la yona. Mohlala: mola o fetang \((2, 3)\) o nang le leralla la 4: \[ y - 3 = 4(x - 2) \] e ka nolofatswang ho \(y = 4x - 5\). d) Sebopeho sa Dintlha tse Pedi Haeba dintlha tse pedi \((x_1, y_1)\) le \((x_2, y_2)\) di tsejwa, equation ya mola e ka fumanwa ho: \[ \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} \] Sebopeho sena se hokahanya ka ho toba dintlha tsohle \((x, y)\) tse moleng o nang le dintlha tse pedi. e) Sebopeho sa ho Thibela Haeba mola o kopana le \(x\)-axis ho \((a, 0)\) le \(y\)-axis ho \((0, b)\), equation ke: \[ \frac{x}{a} + \frac{y}{b} = 1 \] Sebopeho sena se thusa ho bona hobane se hatisa dintlha tsa ho kopana le dilepe. 4. Ho Fumana Tekanyo ea Mola o Otlolohileng Jeomethri ea tlhahlobo, potso e reng "fumana equation ea mola" hangata e hlaha ho latela tlhaiso-leseling e itseng. Mona ke maemo a tloaelehileng: a) Ha ho fanoe ka Leralla le Intercept \(y\) Haeba leralla \(m\) le intercept \(c\) li tsejoa, sebelisa ka ho toba \(y = mx + c\). Mohlala: gradient \(-2\), intercept \(y\) = 3: \[ y = -2x + 3 \] b) Ha ho fanoe ka Lintlha tse peli Mohlala, ha ho fanoe ka \((1, 2)\) le \((3, 6)\). Leralla: \[ m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2 \] Sebelisa ntlha \((1, 2)\):
\[ y - 2 = 2(x - 1) \Motsu o Le letona y = 2x \] c) Methalo e bapileng kapa e otlolohileng - Methalo e bapileng: leralla le le leng. - Methalo e otlolohileng: leralla le fapaneng le lebe (haeba ho hlokahala), ke hore \(m_2 = -\frac{1}{m_1}\). Mohlala: mola \(y = 3x + 1\) o na le leralla la 3. Mothalo o otlolohileng ho wona o na le leralla la \(-\frac{1}{3}\). Haeba mola o otlolohileng o feta \((0, 2)\): \[ y - 2 = -\frac{1}{3}(x - 0) \Motsu o Le letona y = -\frac{1}{3}x + 2 \] 5. Ditshebediso ho Jeometri Tekanyo ya mola o otlolohileng ha e bohlokwa feela ho algebra, empa hape e na le thuso haholo ho jeometri: 1. Ho fumana ntlha ya moo mela e mmedi e kopanang teng. Ntlha ea ho kopana e fumanoa ka ho rarolla tsamaiso ea li-equation. 2. Ho bala sebaka ho tloha ntlheng ho ea mola. Ka sebopeho se akaretsang \(Ax + By + C = 0\), sebaka ho tloha ntlheng \((x_0, y_0)\) ho ea mola ke: \[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \] 3. Ho sekaseka lipalo tsa sefofane. Mahlakore a khutlotharo, sekwere, kapa setšoantšo se seng se ka hlalosoa e le mela, kahoo thepa ea setšoantšo e ka ithutoa ka equation ea mola. 4. Ho fumana bisector le bophahamo ba khutlotharo. Bophahamo bo otlolohile lehlakoreng le fanoeng, ha bisector ea angle e na le melao e khethehileng, eo kaofela e ka baloang ho sebelisoa leralla. 6. Qetello Equation ea mola o otlolohileng ke sesebelisoa sa bohlokoa ho jiometri ea tlhahlobo bakeng sa ho emela mela sefofaneng sa coordinate. Ka ho utloisisa leralla le mefuta e fapaneng ea equation—joalo ka \(y = mx + c\), \(Ax + By + C = 0\), foromo ea ntlha-lekhalo, foromo ea lintlha tse peli, le foromo ea axis-intercept—re ka hlalosa le ho sekaseka mela habonolo. Bokhoni bona bo thusa haholo bakeng sa ho rarolla mathata a jeometri a kang ho fumana lintlha tsa moo ho kopanang teng, ho bala libaka, le ho hlahloba hore na mela e bapile kapa e otlolohile. Qetellong, khopolo ena e bonolo e fana ka borokho ba bohlokoa pakeng tsa jeometri ea pono le algebra e hlophisitsoeng.
