The perimeter of a parallelogram whose sides are represented by the lines `x+2y+3=0`,
`3x+4y-5=0,2x+4y+5=0 and 3x+4y-10=0` is equal to

Options

A. `(5)/(2)+5sqrt5` units
B. `5+4sqrt5` units
C. `5+(5)/(2)sqrt5` units
D. `(5+5sqrt5)/(2)` units

Answer

To find the perimeter of the parallelogram formed by the given lines, we will follow these steps:

### Step 1: Identify the lines and their slopes
The lines are:
1. \( x + 2y + 3 = 0 \)
2. \( 3x + 4y - 5 = 0 \)
3. \( 2x + 4y + 5 = 0 \)
4. \( 3x + 4y - 10 = 0 \)

To determine if the lines are parallel, we need to find their slopes.

For the line \( ax + by + c = 0 \), the slope \( m \) is given by \( m = -\frac{a}{b} \).

Calculating the slopes:
- For \( x + 2y + 3 = 0 \):
\[ m_1 = -\frac{1}{2} \]
- For \( 3x + 4y - 5 = 0 \):
\[ m_2 = -\frac{3}{4} \]
- For \( 2x + 4y + 5 = 0 \):
\[ m_3 = -\frac{1}{2} \]
- For \( 3x + 4y - 10 = 0 \):
\[ m_4 = -\frac{3}{4} \]

### Step 2: Identify pairs of parallel lines
From the slopes calculated:
- Lines 1 and 3 are parallel (both have slope \(-\frac{1}{2}\)).
- Lines 2 and 4 are parallel (both have slope \(-\frac{3}{4}\)).

### Step 3: Calculate the distance between the parallel lines
The distance \( d \) between two parallel lines \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) is given by:
\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]

#### Distance between lines 1 and 3:
1. \( C_1 = 3 \) (from line 1)
2. \( C_2 = 5 \) (from line 3)

Using the formula:
\[ d_1 = \frac{|5 - 3|}{\sqrt{1^2 + 2^2}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5} \]

#### Distance between lines 2 and 4:
1. \( C_1 = -5 \) (from line 2)
2. \( C_2 = -10 \) (from line 4)

Using the formula:
\[ d_2 = \frac{|-10 + 5|}{\sqrt{3^2 + 4^2}} = \frac{5}{5} = 1 \]

### Step 4: Calculate the perimeter of the parallelogram
The perimeter \( P \) of a parallelogram is given by:
\[ P = 2(d_1 + d_2) \]
Substituting the values:
\[ P = 2\left(\frac{2\sqrt{5}}{5} + 1\right) = 2\left(\frac{2\sqrt{5}}{5} + \frac{5}{5}\right) = 2\left(\frac{2\sqrt{5} + 5}{5}\right) = \frac{4\sqrt{5} + 10}{5} \]

### Final Answer
The perimeter of the parallelogram is:
\[ \frac{4\sqrt{5} + 10}{5} \]