Question

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.

• A. \( \frac{1}{7} \)
• B. \( \frac{1}{3} \)
• C. \( \frac{3}{7} \)
• D. \( \frac{3}{5} \)

Hint:

In a regular polygon, isosceles triangles are formed when the vertices are symmetrically placed. Count the number of such cases and divide by the total number of ways to choose the vertices.

Answer 1

The Correct Option is D

To form an isosceles triangle, we need to choose two vertices that are equidistant from each other. In a regular 7-sided polygon, there are 7 vertices, and we need to choose 3 vertices. The total number of ways to choose 3 vertices from 7 is: \[ \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \] For the vertices to form an isosceles triangle, the chosen vertices must be symmetrically placed. This can occur in 3 distinct ways. Thus, the probability is: \[ \frac{3}{35} = \frac{3}{5} \] So, the correct answer is (D) \( \frac{3}{5} \).