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Ensvinklede trekanter – Sådan bliver du bedre!

Formålet med denne artikel er, at gøre den studerende i stand til at løse eksamensopgaver i ensvinklede trekanter, til den skriftlige matematik eksamen på niveau C. Vi har kigget tilbage på gamle eksamensopgaver de sidste ti år, som vedrører ensvinklede trekanter. Tidligere eksamensopgaver i ensvinklede trekanter har optrådt i to varianter. En variant hvor man skal beregne en sidelængde, når man ”går” fra en lille trekant til en stor trekant. Den anden variant går på man skal beregne en sidelængde, når man ”går” fra en stor trekant til en lille trekant. Man skal i princippet finde størrelsesforholdet mellem de to trekanter, og herefter gange med en sidelængde for, at finde den anden sidelængde i den ensvinklede trekant. Vi vil herunder gennemgå, hvordan en eksamensopgave skal gribes an, hvis man skal løse en opgave i ensvinklede trekanter.

Eksempel på en eksamensopgave og hvordan den skal løses.

Figuren herunder viser to ensvinklede trekanter. Trekant ABC og trekant DEF samt nogle af trekanternes mål.

Ensvinklede trekanter 1

Bestem længden af DE i trekant DEF og bestem længden af BC i trekant ABC.

Når man skal finde sidelængder i ensvinklede trekanter, skal man i princippet finde målestoksforholdet. Målestoksforholdet kan findes, hvis man kender målene på to ens sidelængder i de ensvinklede trekanter. Hvis man ser på de to trekanter i opgaven, så får man oplyst målene på to ens sidelængder, nemlig længden AC = 4 i trekant ABC og længden DF = 6 i trekant DEF (Se nedenstående markering med rød).

Ensvinklede trekanter 2

 

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Bestemmelse af sidelængden DE i trekant DEF

Vi ønsker først at finde målestoksforholdet, når man går fra den lille trekant ABC til den store trekant DEF, for at kunne bestemme sidelængden DE. Vi beregner målestoksforholdet;

Ensvinklede trekanter 5

Da vi nu kender målestoksforholdet, når man går fra trekant ABC til trekant DEF kan sidelængden DE bestemmes. Det gøres ved gange det beregnede målestoksforhold med den oplyste sidelængde i trekant ABC (sidelængden AB) som svare til den samme sidelængde som vi ønsker at finde nemlig sidelængden DE. Se den røde markering i figuren herunder;

Ensvinklede trekanter 3

Vi beregner nu sidelængden DE i trekant DEF:

Ensvinklede trekanter 6

 

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Bestemmelse af sidelængden BC i trekant ABC

For at kunne bestemme sidelængden BC i trekant ABC, så skal vi kende målestoksforholdet, når man går fra den store trekant DEF til den lille trekant ABC. Vi bestemmer målestoksforholdet;

Ensvinklede trekanter 7

Vi har nu bestemt målestoksforholdet, når man går fra trekant DEF til trekant ABC. Vi kan derfor bestemme sidelængden BC. Det gøres ved gange det beregnede målestoksforhold med den oplyste sidelængde i trekant DEF (sidelængden EF) som svare til den samme sidelængde som vi ønsker at finde nemlig sidelængden BC. Se den røde markering i figuren herunder;

Ensvinklede trekanter 4

Vi beregner nu sidelængden BC i trekant ABC:

Ensvinklede trekanter 8

Har du lært fremgangsmåden ved beregning af sidelængder i ensvinklede trekanter?

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