Find the slope of a line, which passes through the origin, and the midpoint of the line segment joining the points `P (0, 4)`and `B (8, 0)`.

Explanation

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

### Step 1: Find the midpoint of the line segment joining points \( P(0, 4) \) and \( B(8, 0) \).

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 \( P \) and \( B \):
- \( x_1 = 0, y_1 = 4 \)
- \( x_2 = 8, y_2 = 0 \)

Calculating the midpoint:
\[ M = \left( \frac{0 + 8}{2}, \frac{4 + 0}{2} \right) = \left( \frac{8}{2}, \frac{4}{2} \right) = (4, 2) \]

### Step 2: Determine the slope of the line passing through the origin and the midpoint \( M(4, 2) \).

The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

In this case, we will use the origin \( (0, 0) \) and the midpoint \( (4, 2) \):
- \( (x_1, y_1) = (0, 0) \)
- \( (x_2, y_2) = (4, 2) \)

Calculating the slope:
\[ m = \frac{2 - 0}{4 - 0} = \frac{2}{4} = \frac{1}{2} \]

### Step 3: Conclusion

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

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