In a regular decagon there are diagonals of distinct sizes. If all the possible diagonals are drawn and you choose any one diagonal at random then what is the probability that it is neither the shortest one nor it is a longest one ?

Options

A. `4//7`
B. `3//7`
C. `6//7`
D. `5//7`

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Calculate the total number of diagonals in a decagon.
The formula for the number of diagonals in a polygon is given by:
\[ \text{Number of diagonals} = \frac{n(n-3)}{2} \]
where \( n \) is the number of sides of the polygon. For a decagon, \( n = 10 \).

Substituting the value of \( n \):
\[ \text{Number of diagonals} = \frac{10(10-3)}{2} = \frac{10 \times 7}{2} = \frac{70}{2} = 35 \]

### Step 2: Identify the shortest diagonals.
In a regular decagon, the shortest diagonals connect vertices that are two apart. For example:
- Diagonal from vertex 1 to vertex 3
- Diagonal from vertex 2 to vertex 4
- ...
- Diagonal from vertex 10 to vertex 2

Since there are 10 vertices, there are 10 shortest diagonals.

### Step 3: Identify the longest diagonals.
The longest diagonals connect vertices that are directly opposite each other. In a decagon, there are 5 pairs of opposite vertices. The longest diagonals are:
- Diagonal from vertex 1 to vertex 6
- Diagonal from vertex 2 to vertex 7
- Diagonal from vertex 3 to vertex 8
- Diagonal from vertex 4 to vertex 9
- Diagonal from vertex 5 to vertex 10

Thus, there are 5 longest diagonals.

### Step 4: Calculate the number of diagonals that are neither the shortest nor the longest.
To find the number of diagonals that are neither the shortest nor the longest, we subtract the number of shortest and longest diagonals from the total number of diagonals:
\[ \text{Diagonals neither shortest nor longest} = \text{Total diagonals} - \text{Shortest diagonals} - \text{Longest diagonals} \]
Substituting the values:
\[ \text{Diagonals neither shortest nor longest} = 35 - 10 - 5 = 20 \]

### Step 5: Calculate the probability.
The probability that a randomly chosen diagonal is neither the shortest nor the longest is given by:
\[ \text{Probability} = \frac{\text{Number of diagonals neither shortest nor longest}}{\text{Total number of diagonals}} = \frac{20}{35} \]
This fraction can be simplified:
\[ \frac{20}{35} = \frac{4}{7} \]

### Final Answer:
The probability that a randomly chosen diagonal is neither the shortest nor the longest is \( \frac{4}{7} \).

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