MathDB
Equilateral Triangles on a Pentagon

Source:

January 4, 2009
rotationsymmetry

Problem Statement

Several figures can be made by attaching two equilateral triangles to the regular pentagon ABCDE ABCDE in two of the five positions shown. How many non-congruent figures can be constructed in this way? [asy]unitsize(2cm);
pair A=dir(306); pair B=dir(234); pair C=dir(162); pair D=dir(90); pair E=dir(18);
draw(A--B--C--D--E--cycle,linewidth(.8pt)); draw(E--rotate(60,D)*E--D--rotate(60,C)*D--C--rotate(60,B)*C--B--rotate(60,A)*B--A--rotate(60,E)*A--cycle,linetype("4 4"));
label("AA",A,SE); label("BB",B,SW); label("CC",C,WNW); label("DD",D,N); label("EE",E,ENE);[/asy]<spanclass=latexbold>(A)</span> 1<spanclass=latexbold>(B)</span> 2<spanclass=latexbold>(C)</span> 3<spanclass=latexbold>(D)</span> 4<spanclass=latexbold>(E)</span> 5 <span class='latex-bold'>(A)</span>\ 1 \qquad <span class='latex-bold'>(B)</span>\ 2 \qquad <span class='latex-bold'>(C)</span>\ 3 \qquad <span class='latex-bold'>(D)</span>\ 4 \qquad <span class='latex-bold'>(E)</span>\ 5