In triangle ABC, the co-ordinates of vertices A, B and C are (4, 7), (-2, 3) and (0, 1) respectively. Find the equation of median through vertex A.
Also, find the equation of the line through vertex B and parallel to AC.

Answer

To solve the problem, we will follow these steps:

### Step 1: Find the midpoint of side BC
The coordinates of points B and C are given as:
- B(-2, 3)
- C(0, 1)

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

Using this formula, we can find the midpoint \( M \) of segment BC:
\[ M = \left( \frac{-2 + 0}{2}, \frac{3 + 1}{2} \right) = \left( \frac{-2}{2}, \frac{4}{2} \right) = (-1, 2) \]

### Step 2: Find the slope of line AM
Now, we need to find the slope of the line connecting point A to the midpoint M. The coordinates of A are (4, 7).

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

For points A(4, 7) and M(-1, 2):
\[ m_{AM} = \frac{2 - 7}{-1 - 4} = \frac{-5}{-5} = 1 \]

### Step 3: Use point-slope form to find the equation of line AM
The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Using point A(4, 7) and the slope \( m = 1 \):
\[ y - 7 = 1(x - 4) \]
Simplifying this gives:
\[ y - 7 = x - 4 \implies y = x + 3 \]

### Step 4: Find the equation of the line through B parallel to AC
Next, we need to find the slope of line AC. The coordinates of C are (0, 1).

Using the slope formula again for points A(4, 7) and C(0, 1):
\[ m_{AC} = \frac{1 - 7}{0 - 4} = \frac{-6}{-4} = \frac{3}{2} \]

Since the line through B should be parallel to AC, it will have the same slope \( m = \frac{3}{2} \).

### Step 5: Use point-slope form to find the equation of the line through B
Using point B(-2, 3) and the slope \( m = \frac{3}{2} \):
\[ y - 3 = \frac{3}{2}(x + 2) \]
Expanding this:
\[ y - 3 = \frac{3}{2}x + 3 \implies y = \frac{3}{2}x + 6 \]

### Final Answers
1. The equation of the median through vertex A is:
\[ y = x + 3 \]
2. The equation of the line through vertex B and parallel to AC is:
\[ y = \frac{3}{2}x + 6 \]