The slopes of the line which passes through the origin, and the mid - point of the line segment joining the points. `P(3, -4)` and `Q(-5, -2)` is

Options

A. 3
B. `-3`
C. `-1`
D. 1

Answer

To find the slope of the line that passes through the origin and the midpoint of the line segment joining the points \( P(3, -4) \) and \( Q(-5, -2) \), we can follow these steps:

### Step 1: Find the Midpoint of the Line Segment
The formula for the midpoint \( M \) of a line segment joining two points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) is given by:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

For the points \( P(3, -4) \) and \( Q(-5, -2) \):
- \( x_1 = 3 \), \( y_1 = -4 \)
- \( x_2 = -5 \), \( y_2 = -2 \)

Substituting these values into the midpoint formula:

\[ M = \left( \frac{3 + (-5)}{2}, \frac{-4 + (-2)}{2} \right) \]

Calculating the x-coordinate:

\[ \frac{3 - 5}{2} = \frac{-2}{2} = -1 \]

Calculating the y-coordinate:

\[ \frac{-4 - 2}{2} = \frac{-6}{2} = -3 \]

Thus, the midpoint \( M \) is:

\[ M(-1, -3) \]

### Step 2: Calculate the Slope of the Line through the Origin and Midpoint
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 need to find the slope of the line passing through the origin \( O(0, 0) \) and the midpoint \( M(-1, -3) \).

Substituting the coordinates into the slope formula:

\[ m = \frac{-3 - 0}{-1 - 0} = \frac{-3}{-1} = 3 \]

### Final Answer
The slope of the line that passes through the origin and the midpoint of the line segment joining the points \( P(3, -4) \) and \( Q(-5, -2) \) is:

\[ \boxed{3} \]