In a regular 15 -sided polygon with all its diagonals drawn, a diagonal is chosen at random. The probability that it is either a shortest diagonal nor a longest diagonal is

Question

In a regular 15 -sided polygon with all its diagonals drawn, a diagonal is chosen at random. The probability that it is either a shortest diagonal nor a longest diagonal is (A) (2/3) (B) (5/6) (C) (8/9) (D) (9/10)

Tardigrade Admin · Accepted Answer

Total diagonals = 15 C 2-15=90 Shortest diagonal = Diagonal connecting ( A 1 A 3, A 2 A 4, ldots) =15 <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/8ea09eed51c9a442fab01c6ab5aa22e3-.png" /> longest diagonal = Diagonal connecting (A1 A8, A1 A9, ldots) =15 Required probability =(90-15-15/90) =(60/90)=(2/3)

Q. In a regular $15$ -sided polygon with all its diagonals drawn, a diagonal is chosen at random. The probability that it is either a shortest diagonal nor a longest diagonal is

$\frac{2}{3}$

$\frac{5}{6}$

$\frac{8}{9}$

$\frac{9}{10}$

Total diagonals $=^{15} C_{2} - 15 = 90$
Shortest diagonal = Diagonal connecting
$(A_{1} A_{3}, A_{2} A_{4}, \dots)$
$= 15$

longest diagonal = Diagonal connecting
$(A_{1} A_{8}, A_{1} A_{9}, \dots)$
$= 15$
Required probability $= \frac{90 - 15 - 15}{90}$
$= \frac{60}{90} = \frac{2}{3}$

Q. In a regular 15 -sided polygon with all its diagonals drawn, a diagonal is chosen at random. The probability that it is either a shortest diagonal nor a longest diagonal is

32​

65​

98​

109​