Find the value of N if point P(-3,N) bisects the line joining points A(12,-1) and B(-18,17) equally.

Options

A. -2
B. 3
C. 6
D. 8

Answer

To find the value of \( N \) such that the point \( P(-3, N) \) bisects the line segment joining points \( A(12, -1) \) and \( B(-18, 17) \), we can use the midpoint formula.

### Step-by-Step Solution:

1. **Understand the Midpoint Formula**:
The midpoint \( M \) of a line segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

2. **Identify the Coordinates**:
Here, we have:
- Point \( A(12, -1) \) where \( x_1 = 12 \) and \( y_1 = -1 \)
- Point \( B(-18, 17) \) where \( x_2 = -18 \) and \( y_2 = 17 \)
- Point \( P(-3, N) \) where we need to find \( N \)

3. **Calculate the Midpoint**:
Using the midpoint formula, we calculate the midpoint \( M \) of the line segment \( AB \):
\[ M = \left( \frac{12 + (-18)}{2}, \frac{-1 + 17}{2} \right) \]

4. **Simplify the Coordinates**:
- For the x-coordinate:
\[ \frac{12 - 18}{2} = \frac{-6}{2} = -3 \]
- For the y-coordinate:
\[ \frac{-1 + 17}{2} = \frac{16}{2} = 8 \]

5. **Set Up the Equation**:
Since point \( P(-3, N) \) is the midpoint, we can equate the y-coordinates:
\[ N = 8 \]

6. **Conclusion**:
Therefore, the value of \( N \) is:
\[ N = 8 \]