Three vertices are chosen randomly from the seven vertices of a regular 7 -sided polygon. The probability that they form the vertices of an isosceles triangle is

Step-by-step solution

The total number of ways to choose 3 vertices out of 7 is given by ⁷C₃ = 7 6 5 3 2 1 = 35 . An isosceles triangle is formed if at least two sides of the triangle are equal. In a regular 7-sided polygon, let the vertices be V₀, V₁, V₂, V₃, V₄, V₅, V₆ . For any vertex chosen as the apex of the isosceles triangle, there are 3 possible pairs of other vertices that form an isosceles triangle with it. For example, if V₀ is the apex, the pairs are (V₁, V₆) , (V₂, V₅) , and (V₃, V₄) . Since there are 7 vertices, and each v