MathDB
Octagons and Triangles

Source: 2018 AMC 8 #23

November 20, 2018
probability2018 AMC 8AMC 8

Problem Statement

From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?
[asy] size(3cm); pair A[]; for (int i=0; i<9; ++i) { A = rotate(22.5+45*i)*(1,0); } filldraw(A[0]--A[1]--A[2]--A[3]--A[4]--A[5]--A[6]--A[7]--cycle,gray,black); for (int i=0; i<8; ++i) { dot(A); } [/asy]
<spanclass=latexbold>(A)</span>27<spanclass=latexbold>(B)</span>542<spanclass=latexbold>(C)</span>1114<spanclass=latexbold>(D)</span>57<spanclass=latexbold>(E)</span>67<span class='latex-bold'>(A) </span> \frac{2}{7} \qquad <span class='latex-bold'>(B) </span> \frac{5}{42} \qquad <span class='latex-bold'>(C) </span> \frac{11}{14} \qquad <span class='latex-bold'>(D) </span> \frac{5}{7} \qquad <span class='latex-bold'>(E) </span> \frac{6}{7}