Given points `A(1, 5), B(-3,7) and C(15,9)`
Find the equation of a line passing through the mid-point of AC and the point B.

Answer

To find the equation of a line passing through the midpoint of points A and C and point B, we will follow these steps:

### Step 1: Find the Midpoint of AC
The coordinates of points A and C are given as:
- A(1, 5)
- C(15, 9)

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

Substituting the coordinates of A and C:
\[ M = \left( \frac{1 + 15}{2}, \frac{5 + 9}{2} \right) = \left( \frac{16}{2}, \frac{14}{2} \right) = (8, 7) \]

### Step 2: Identify the Coordinates of Point B
The coordinates of point B are given as:
- B(-3, 7)

### Step 3: Use the Two Points to Find the Equation of the Line
We now have two points through which the line passes:
- Midpoint M(8, 7)
- Point B(-3, 7)

We can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
where \( m \) is the slope of the line and \( (x_1, y_1) \) is a point on the line.

### Step 4: Calculate the Slope (m)
The slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using points M(8, 7) and B(-3, 7):
\[ m = \frac{7 - 7}{-3 - 8} = \frac{0}{-11} = 0 \]

### Step 5: Substitute into the Point-Slope Formula
Since the slope \( m \) is 0, the equation simplifies to:
\[ y - 7 = 0 \]

### Step 6: Final Equation of the Line
Thus, the equation of the line is:
\[ y = 7 \]

### Summary of the Steps:
1. Find the midpoint of points A and C.
2. Identify the coordinates of point B.
3. Calculate the slope using the two points.
4. Substitute into the point-slope form.
5. Simplify to get the final equation.