The slope of a line which passes through the origin the mid-point of the line segment joining the points `(0,-4)` and `(8,0)` is

Options

A. `(1)/(2)`
B. `-(1)/(2)`
C. `1`
D. `2`

Answer

To find the slope of the line that passes through the origin and the midpoint of the line segment joining the points (0, -4) and (8, 0), we can follow these steps:

### Step 1: Identify the given points
The points given are:
- Point A: (0, -4)
- Point B: (8, 0)

### Step 2: Calculate the midpoint of the line segment
The formula for the midpoint \( M \) of a line segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Substituting the coordinates of points A and B:
\[ M = \left( \frac{0 + 8}{2}, \frac{-4 + 0}{2} \right) = \left( \frac{8}{2}, \frac{-4}{2} \right) = (4, -2) \]

### Step 3: Determine the slope of the line
The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, we will use the origin (0, 0) as one point and the midpoint (4, -2) as the other point:
- Point 1 (origin): \( (0, 0) \) → \( x_1 = 0, y_1 = 0 \)
- Point 2 (midpoint): \( (4, -2) \) → \( x_2 = 4, y_2 = -2 \)

Substituting these values into the slope formula:
\[ m = \frac{-2 - 0}{4 - 0} = \frac{-2}{4} = -\frac{1}{2} \]

### Conclusion
The slope of the line that passes through the origin and the midpoint of the line segment joining the points (0, -4) and (8, 0) is:
\[ \boxed{-\frac{1}{2}} \]