Pfungwa huru dzeEuclidean Geometry
Euclidean geometry ibazi remasvomhu rinoongorora chimiro, saizi, nzvimbo, uye hunhu hwenzvimbo zvichibva papfungwa dzakagadzirwa naEuclid (c. muzana remakore rechitatu BC) mubasa rake guru, Elements. Kwemazana emakore, geometry iyi yave iri hwaro hukuru hwekunzwisisa nzvimbo ine mativi maviri (plane) uye mativi matatu (space) sezvatinosangana nayo muhupenyu hwezuva nezuva. Patinodhirowa mutsetse wakatwasuka neruler, kuyera makona etriangle, kana kuverenga nzvimbo ye rectangle, tiri kushandisa misimboti yeEuclidean geometry. Chinyorwa chino chinokurukura pfungwa huru dzeEuclidean geometry, zvinhu zvayo zvekutanga, ma axioms, uye mamwe ma theorems akakosha anoumba hwaro hwayo.
1. Mapoinzi, Mitsetse, uye Mapurani: Zvinhu Zvinokosha
Geometry yeEuclidean inovakwa nezvinhu zvitatu zvikuru: mapoinzi, mitsetse, uye mapurani.
1. Poindi chinhu chiri nyore chinongoratidza nzvimbo chete uye chisina miganhu (hapana kureba, upamhi, kana kukwirira). Poindi dzinowanzo fananidzirwa nemabhii makuru akadai saA, B, kana C.
2. Mutsetse iboka remapoinzi anotambanuka mumativi maviri uye ane divi rimwe chete, kureva kureba. Mu geometry yakanaka, mutsetse hauna ukobvu. Mutsetse unogona kutsanangurwa nemapoinzi maviri akasiyana, semuenzaniso, mutsetse unopfuura nemuA neB unonzi mutsetse AB.
3. Ndege inzvimbo yakati sandara inotambanuka isingaperi kumativi ese, ine mativi maviri (kureba nehupamhi), uye isina ukobvu. Ndege inogona kutsanangurwa nezvikamu zvitatu zvisiri mumutsara wakatwasuka.
Kunyange zvazvo mifananidzo iri pamapepa ichiita seine ukobvu uye mapurani anoita seane miganhu, muEuclidean mathematical concept, zvese izvi ipfungwa dzakanaka.
2. Mafungiro aEuclid neBasa reAxioms
Hunhu hweEuclidean geometry hunhu hwayo hwekutsvaga: kutanga kubva pamashoko ekutanga anogamuchirwa pasina humbowo (axioms kana postulates), wozoawana mudzidziso kuburikidza nehumbowo hwepfungwa.
Euclid akagadzira mapoulates mashanu ane mukurumbira. Muchimiro chemazuva ano chakapfupika, mapoulates aya anogona kunzwisiswa se:
1. Mapoinzi maviri akasiyana anosarudza mutsetse mumwe chete wakatwasuka.
2. Chikamu chemutsetse chinogona kuwedzerwa nguva dzose kuti chigadzire mutsetse wakatwasuka.
3. Nenzvimbo yakati uye radius, denderedzwa rinogona kugadzirwa.
4. Makona ese ekurudyi akaenzana.
5. Parallel postulate: Kana mutsetse ukayambuka mimwe mitsetse miviri zvekuti huwandu hwemakona emukati kune rumwe rutivi huri pasi pe180°, ipapo mitsetse miviri ichayambuka kune rumwe rutivi kana yakatambanudzwa.
Pfungwa yechishanu iyi ndiyo inonyanya kukakavadzana, sezvo isingaratidzike se "yakareruka" kupfuura dzimwe ina. Kuedza kuiratidza kubva kune dzimwe pfungwa kwakakundikana kwemazana emakore, zvichizopedzisira zvagadzira nzira yekutanga kwechimiro cheEuclidean. Asi chero bedzi pfungwa yechishanu ichigamuchirwa, tinoramba tiri mukati mehurongwa hweEuclidean.
3. Pfungwa yeMitsetse Yakafanana neYakamira
MuEuclidean geometry, mitsara miviri iri mudenderedzwa inonzi yakafanana kana isingambosangani kunyangwe yakatambanudzwa nekusingaperi. Chinhu chakakosha: kuburikidza nepoindi iri kunze kwemutsetse, pane mutsetse mumwe chete wakaenzana nemutsetse iwoyo (zvichienderana nepfungwa yakafanana).
Zvichakadaro, mitsetse miviri inonzi yakatwasuka kana ikagovana pakona ye90°. Pfungwa yekutwasuka ndiyo hwaro hwakakosha hwekugadzira masisitimu ekubatanidza, kugadzira manhamba epakati, uye kuyera makona.
4. Makona uye Kuyera Kwawo
Kona inoumbwa nemwaranzi miviri inosangana panzvimbo yekutanga (vertex). Makona anoyerwa nemadhigirii (°) kana kuti radians. MuEuclidean geometry, mamwe emhando dzemakona dzinonyanya kutaurwa nezvadzo anosanganisira:
– Akone yeAcute: 0° < angle < 90° - Akone yeKurudyi: angle = 90° - Akone yeObtuse: 90° < angle < 180° - Akone yakatwasuka: angle = 180° Hukama huripo pakati pemakona hwakakoshawo, semuenzaniso, angles dzekuwedzera (sum 180°), angles dzekuwedzera (sum 90°), uye angles dzakapesana (dzakaenzana). 5. Maumbirwo ePlane: Triangles, Quadrilaterals, uye Denderedzwa a. Triangles Triangle imhando yeplane yakaganhurirwa nemativi matatu. MuEuclidean geometry, triangle ine hunhu hwakakosha: huwandu hweangles mutriangle i180°. Izvi zvakasiyana mu non-Euclidean geometry. Matatu anogona kupatsanurwa zvichienderana nemativi: - Equilateral: mativi ese matatu akaenzana - Isosceles: mativi maviri akaenzana - Chero: mativi ese akasiyana Uye zvichibva pamakona: - Acute, right, obtuse Theorem inozivikanwa mumatatu matatu iPythagorean Theorem, iyo inoshanda kune matatu matatu ekurudyi: \(a^2 + b^2 = c^2\) apo \(c\) iri hypotenuse. b. Quadrilaterals Quadrilateral ine mativi mana. Mamwe maquadrilateral akakosha: - Square: mativi ese akaenzana, makona ese ari 90° - Rectangle: makona ari 90°, mativi akatarisana akaenzana - Parallelogram: mativi akatarisana akaenzana uye akaenzana - Rhombus: mativi ese akaenzana - Trapezoid: ine peya imwe yemativi akaenzana Imwe neimwe ine hunhu hwayo hwakasiyana hwemakona nemadiagonal, izvo zvinogona kuratidzwa nenzira yeEuclidean. c. Denderedzwa Denderedzwa iboka remapoinzi akaenzana kubva panzvimbo yepakati. Pfungwa dzakakosha dziri mumadenderedzwa dzinosanganisira: - Radius (r), dhayamita (2r) - Kutenderera: \(K = 2\pi r\) - Nzvimbo: \(L = \pi r^2\) Pamusoro pezvo, kune pfungwa dze arcs, chords, sectors, segments, pamwe ne central angles ne circumference angles. 6. Kufanana uye Kubatana Maumbirwo maviri anonzi akaenzana kana chimiro chawo nehukuru zvakafanana (zvinogona kuiswa kuburikidza nekushandura, kutenderera, kana kufungisisa). Semuenzaniso, matriangle maviri akabatana ane mativi nemakona akafanana.
Maumbirwo maviri anonzi akafanana kana aine chimiro chakafanana asi anogona kusiyana muhukuru; chiyero chemativi anoenderana hachichinji. Kufanana kwakakosha zvikuru mukugadzira mapa, kudhirowa zviyero, magadzirirwo, uye kuyera zvisina kunanga (semuenzaniso kuyera kureba kwemuti uchishandisa mumvuri wawo). 7. Kuchinja kweGeometric muEuclidean Space Euclidean geometry inodzidzawo shanduko dzinochengetedza zvimwe zvinhu. Kuchinja kwekutanga kunosanganisira: - Kushandura (shift): kufambisa mapoinzi ese nevector imwechete - Kutenderera (kutenderera): kutenderera chimiro chakatenderedza nzvimbo yepakati yekutenderera - Kufungisisa (girazi): kuratidza chimiro pamutsetse (mundege) kana ndege (munzvimbo) - Kukura (zoom in/out): kushandura saizi nechiyero chinhu Kuchinja kwakadai sekushandura, kutenderera, uye kufungisisa kuchengetedza madaro nemakona (isometries), nepo kuwedzera kuchichengetedza chimiro asi kuchichinja saizi. 8. Nei Euclidean Geometry Yakakosha? Euclidean geometry haisi yakakosha chete sedzidziso yemasvomhu, asiwo sechishandiso chinoshanda muminda yakasiyana-siyana: mainjiniya ehurumende, magadzirirwo ezvivakwa, dhizaini yechigadzirwa, mifananidzo yekombuta, mapikicha, uye kunyange fizikisi yekare. Nzvimbo dzatinoona se "dzakajairika" pachikero chezuva nezuva dzinogona kuenzaniswa zvakanaka neEuclidean geometry. Kunyangwe pachikero chepasi rose kana mudzidziso yerelativity, nzvimbo inogona kukombama (isiri yeEuclidean), Euclidean geometry inoramba iri nyore kunzwisisa uye inonyanya kushandiswa kutanga hwaro. Mhedziso Pfungwa huru dzeEuclidean geometry dzinotanga nezvinhu zvakakosha - mapoinzi, mitsara, uye mapurani - uye dzobva dzakura kuburikidza nezviratidzo uye humbowo hunoisa dzidziso dzakakosha nezvemakona, mitsetse yakafanana, uye nhamba dzakasiyana-siyana dzepuraneti dzakadai setriangles, quadrilaterals, uye denderedzwa. Nehurongwa hwayo hwakarongeka uye hwakarongeka, Euclidean geometry ndeimwe yebudiriro huru dzepfungwa munhoroondo yemasvomhu, pamwe nechishandiso chinoshanda chinoramba chakakosha nanhasi. Kunzwisisa zvinhu zvepakutanga idanho rekutanga rakasimba rekudzidza masvomhu epamusoro-soro, kusanganisira analytic geometry, trigonometry, uye non-Euclidean geometry.