Two sides of a square lie on the lines x + y = 1 and x + y + 2 = 0. What is its area?

Options

A. `(11)/(2)`
B. (9)/(2)`
C. 5
D. 4

Answer

To find the area of the square whose two sides lie on the lines \( x + y = 1 \) and \( x + y + 2 = 0 \), we will follow these steps:

### Step 1: Identify the equations of the lines
The given lines are:
1. \( x + y = 1 \)
2. \( x + y + 2 = 0 \) which can be rewritten as \( x + y = -2 \)

### Step 2: Determine the distance between the two parallel lines
The formula for the perpendicular distance \( d \) between two parallel lines of the form \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) is given by:
\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]
For our lines, we can rewrite them as:
1. \( x + y - 1 = 0 \) (where \( C_1 = -1 \))
2. \( x + y + 2 = 0 \) (where \( C_2 = 2 \))

Thus, we have:
- \( A = 1 \)
- \( B = 1 \)
- \( C_1 = -1 \)
- \( C_2 = 2 \)

Now, substituting these values into the distance formula:
\[ d = \frac{|2 - (-1)|}{\sqrt{1^2 + 1^2}} = \frac{|2 + 1|}{\sqrt{1 + 1}} = \frac{3}{\sqrt{2}} \]

### Step 3: Calculate the side length of the square
Since the distance \( d \) between the two lines is equal to the length of the diagonal of the square, we can find the side length \( s \) of the square using the relationship between the diagonal \( d \) and the side length \( s \):
\[ d = s\sqrt{2} \]
Thus, we can solve for \( s \):
\[ s = \frac{d}{\sqrt{2}} = \frac{3/\sqrt{2}}{\sqrt{2}} = \frac{3}{2} \]

### Step 4: Calculate the area of the square
The area \( A \) of the square is given by:
\[ A = s^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]

### Step 5: Final result
The area of the square is:
\[ \frac{9}{4} \text{ square units} \]