Q89 / 180 · COMEDK 2022 · Mathematics
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$\dfrac{1}{7}$
- B$\dfrac{1}{3}$
- C$\dfrac{3}{7}$
- Correct answerD$\dfrac{3}{5}$
Solution
The total number of ways to choose 3 vertices out of 7 is given by $^{7}C_{3} = \dfrac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$.
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 $V_{0}, V_{1}, V_{2}, V_{3}, V_{4}, V_{5}, V_{6}$.
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 $V_{0}$ is the apex, the pairs are $(V_{1}, V_{6})$, $(V_{2}, V_{5})$, and $(V_{3}, V_{4})$.
Since there are 7 vertices, and each vertex can serve as the apex for 3 distinct isosceles triangles, the total number of isosceles triangles is $7 \times 3 = 21$.
The probability is the ratio of the number of isosceles triangles to the total number of triangles: $P = \dfrac{21}{35} = \dfrac{3}{5}$.
Answer: $\dfrac{3}{5}$
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 $V_{0}, V_{1}, V_{2}, V_{3}, V_{4}, V_{5}, V_{6}$.
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 $V_{0}$ is the apex, the pairs are $(V_{1}, V_{6})$, $(V_{2}, V_{5})$, and $(V_{3}, V_{4})$.
Since there are 7 vertices, and each vertex can serve as the apex for 3 distinct isosceles triangles, the total number of isosceles triangles is $7 \times 3 = 21$.
The probability is the ratio of the number of isosceles triangles to the total number of triangles: $P = \dfrac{21}{35} = \dfrac{3}{5}$.
Answer: $\dfrac{3}{5}$
Hence, the correct answer is option: 'D'.