Stereometrie - Gymnázium Rožnov pod Radhoštěm

Stereometrie - Gymnázium Rožnov pod Radhoštěm

STEREOMETRIE Vzjemn poloha t rovin Mgr. Jakub Nmec VY_32_INOVACE_M3r0107 VZJEMN POLOHA T ROVIN V tto lekci si ukeme, jak vzjemn polohy mohou mt ti roviny. Tchto poloh je pt: Ti roviny jsou po dvou rovnobn nemaj dn spolen bod.

Dv roviny jsou rovnobn a tet je k nim rznobn existuj dv rovnobn pmky (prsenice). Ti roviny jsou po dvou rznobn a jejich prsenice splynou v jednu pmku tzv. svazek rovin. Ti roviny jsou po dvou rznobn a maj ti rzn rovnobn prsenice. Ti roviny jsou po dvou rznobn a jejich rzn prsenice se protnou v jednom bod tzv. trs rovin. V nsledujcch snmcch si kadou vzjemnou polohu ukeme.

TI NAVZJEM ROVNOBN ROVINY V krychli ABCDEFGH mjme roviny ADE, BCF a KLM, kde body K, L a M jsou po ad stedy hran AB, CD a GH. Dokzat

vzjemnou rovnobnost tchto t rovin je jednoduch cvien (vyuit rovnobnosti rznobnch pmek v rovinch). Zde vyznaeny rznobky v rovinch, kter jsou

navzjem rovnobn. V krychli ABCDEFGH mjme roviny BCE, KLM a RST, kde body K, L, M, R, S a T jsou po ad stedy hran AB, CD, AE, BF, CG a EF. Dkaz vzjemn rovnobnosti je

zaloen opt na rovnobnosti rznobnch pmek. Zde vyznaeny rznobky v rovinch, kter jsou navzjem rovnobn. DV ROVNOBN ROVINY A TET

RZNOBN V krychli ABCDEFGH mjme roviny ABC, BCF a EFG. Je zejm, e podstavy jsou rovnobn roviny. Pmky BC a FG, kter jsou prsenicemi podstav z bon

stnou jsou rovnobn, co vyplv z vlastnost krychle. V krychli ABCDEFGH mjme roviny ABL, KGH a CEF, kde body K a L jsou po ad stedy hran AE a CG.

Rovnobnost rovin ABL a KGH je snadno dokazateln (opt pomoc rznobek v rovinch, kter jsou navzjem rovnobn). Zelenou a rovou barvou jsou vyznaeny

prsenice, kter jsou rovnobn. TI NAVZJEM RZNOBN ROVINY S JEDNOU PRSENIC V krychli ABCDEFGH mjme roviny ABC, BCE a BCF. Ji z pojmenovn rovin je jasn, e tyto navzjem rznobn

roviny maj dva spolen body, co je dostaujc k uren prsenice (pmku uruj dva rzn body). Spolen prsenice pro roviny ABC, BCE a BCF je pmka BC.

V krychli ABCDEFGH mjme roviny BCE, KLM a RST, kde body K, L, M, R, S a T jsou po ad stedy hran AB, EF, GH, AE, BF a CG. Je patrn, e roviny jsou navzjem rznobn.

Vechny ti roviny se protnaj ve stedu pedn (P) a zadn stny (Q), co lze dokzat pomoc vlastnosti krychle. Prseky P a Q nm jednoznan uruj prsenici

PQ danch t rovin. TI NAVZJEM RZNOBN ROVINY SE TEMI RZNMI PRSENICEMI V krychli ABCDEFGH mjme roviny ABC, ABL a CDK, kde body K a L jsou po ad stedy hran EH a

FG. Opt je zcela patrn, e roviny jsou navzjem rznobn (nememe najt dv rznobky v rovin, kter by mly sv rovnobky v ostatnch rovinch). Spolen body

po dvou rznobnch rovin uruj hledan prsenice. Dv z nich najdeme v doln podstav, kde se protnaj roviny ABL a CDK s rovinou ABC. Tet prsenice je urena body K a L, kter nle rovinm ABL

a CDK souasn. V krychli ABCDEFGH mjme roviny ACE, DHK a DHL, kde body K a L jsou po ad stedy hran AB a BC. Vidme, e jednotliv roviny jsou navzjem rznobn.

Roviny se protnaj vdy v horn a doln podstav. Spojenm pslunch bod zskme hledan prsenice. TI NAVZJEM RZNOBN ROVINY S JEDNM SPOLENM BODEM V krychli

ABCDEFGH mjme roviny BCG, CDG a EFG. Ji s pojmenovn rovin je zejm, e vechny roviny maj alespo jeden spolen bod. Vzhledem k tomu, e roviny jsou navzjem rznobn, bude tento bod

zrove jedin a uren temi rznobnmi prsenicemi rovin. Zadan roviny maj prsenice CG, FG a HG. Vechny se protnaj v bod G. V krychli

ABCDEFGH mjme roviny KLM, RST a XYZ, kde body K, L, M, R, S, T, X, Y a Z jsou po ad stedy hran AB, EF, GH, AE, BF, CG, BC, FG a EH. Prseky po dvou rznch danch rovin jsou vdy ve stedu

protilehlch stn, co vychz z vlastnost krychle. Spojenm pslunch bod zskme ti rznobn prsenice, kter se protnaj v jednom bod (P).

KOL ZVREM V krychli ABCDEFGH uri vzjemnou polohu t rovin urench body: a) BGE, ACH a ACG b) ADE, BCF a KLM, kde body K, L a M jsou po ad stedy hran AB, CD a EF c) ACE, BDH a RST, kde body R, S a T jsou po ad stedy hran AE, BF a CG d) ACG, BDH a XYZ, kde body X, Y a Z jsou po ad stedy hran BC, FG a EH e) ACE, BDH a BCF.

ZDROJE Literatura: POMYKALOV, Eva. Matematika pro gymnzia - Stereometrie. 1. vydn. Praha: Prometheus, 1995, 223 s. ISBN 80-7196-004-7. Obrzky byly vytvoeny v programu Malovn.

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