Find the equations of the diagonals of a rectangle whose sides are x + 1 = 0, x - 4 = 0, y + 1 = 0 and y - 2 = 0.

Answer

To find the equations of the diagonals of a rectangle defined by the lines \(x + 1 = 0\), \(x - 4 = 0\), \(y + 1 = 0\), and \(y - 2 = 0\), we can follow these steps:

### Step 1: Identify the vertices of the rectangle
The given equations represent the sides of the rectangle:
- \(x + 1 = 0\) gives \(x = -1\)
- \(x - 4 = 0\) gives \(x = 4\)
- \(y + 1 = 0\) gives \(y = -1\)
- \(y - 2 = 0\) gives \(y = 2\)

The vertices of the rectangle can be determined by combining these x and y values:
- Vertex A: \((-1, -1)\)
- Vertex B: \((-1, 2)\)
- Vertex C: \((4, -1)\)
- Vertex D: \((4, 2)\)

### Step 2: Find the equations of the diagonals
The diagonals of the rectangle are \(AC\) and \(BD\).

#### Diagonal AC:
1. **Identify the coordinates**:
- Point A: \((-1, -1)\)
- Point C: \((4, -1)\)

2. **Calculate the slope (m)**:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - (-1)}{4 - (-1)} = \frac{0}{5} = 0 \]
Since the slope is 0, the line is horizontal.

3. **Equation of line AC**:
Since it is a horizontal line at \(y = -1\), the equation is:
\[ y = -1 \]

#### Diagonal BD:
1. **Identify the coordinates**:
- Point B: \((-1, 2)\)
- Point D: \((4, 2)\)

2. **Calculate the slope (m)**:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 2}{4 - (-1)} = \frac{0}{5} = 0 \]
Again, the slope is 0, indicating a horizontal line.

3. **Equation of line BD**:
Since it is also a horizontal line at \(y = 2\), the equation is:
\[ y = 2 \]

### Final Equations of the Diagonals:
- Diagonal AC: \(y = -1\)
- Diagonal BD: \(y = 2\)