The number of pairs of diagonals of a regular polygon of `10` sides that are parallel are

Options

A. `45`
B. `56`
C. `22`
D. `64`

Explanation

To find the number of pairs of diagonals of a regular polygon with 10 sides that are parallel, we can follow these steps:

### Step 1: Understand the Structure of the Polygon
A regular polygon with 10 sides (decagon) has vertices labeled as A, B, C, D, E, F, G, H, I, and J. Each vertex connects to other vertices to form sides and diagonals.

**Hint:** Visualize the polygon and label its vertices to keep track of the connections.

### Step 2: Identify Parallel Diagonals
For any given diagonal in a regular polygon, there are diagonals that are parallel to it. A diagonal connects two non-adjacent vertices. In a decagon, each vertex connects to 7 other vertices (excluding itself and its two adjacent vertices).

**Hint:** Remember that each diagonal can be paired with another diagonal that is opposite to it in the polygon.

### Step 3: Count the Diagonals
The formula to calculate the total number of diagonals \(D\) in a polygon with \(n\) sides is given by:
\[ D = \frac{n(n-3)}{2} \]
For a decagon (\(n = 10\)):
\[ D = \frac{10(10-3)}{2} = \frac{10 \times 7}{2} = 35 \]

**Hint:** Use the formula to calculate the total diagonals before focusing on pairs.

### Step 4: Determine Pairs of Parallel Diagonals
To find pairs of parallel diagonals, we can fix one diagonal and count how many diagonals are parallel to it. Each diagonal has exactly one parallel counterpart.

1. **Fix a diagonal:** For example, if we fix diagonal AB, the parallel diagonals can be:
- CD
- EF
- GH
- IJ

Thus, for each diagonal, there are 4 parallel diagonals.

**Hint:** Each diagonal will have a specific set of diagonals that are parallel to it based on their arrangement.

### Step 5: Calculate the Total Number of Pairs
Since each diagonal can form a pair with its parallel counterpart, we can calculate the number of pairs of diagonals:
- Each diagonal can form a pair with 4 other diagonals.
- Since there are 35 diagonals, we can calculate the number of pairs of diagonals as follows:
\[ \text{Total pairs} = \frac{4 \times \text{Number of diagonals}}{2} = \frac{4 \times 35}{2} = 70 \]

**Hint:** Remember to divide by 2 to avoid double counting pairs.

### Step 6: Final Calculation
After considering the arrangement and counting, we find that the total number of pairs of parallel diagonals in a regular polygon of 10 sides is:

\[ \text{Total pairs of parallel diagonals} = 70 \]

**Final Answer:** The number of pairs of diagonals of a regular polygon of 10 sides that are parallel is **70**.