`A(-3, 1), B(4,4) and C(1, -2)` are the vertices of a triangle ABC. Find:
the equation of median BD.

Answer

To find the equation of the median BD of triangle ABC with vertices A(-3, 1), B(4, 4), and C(1, -2), we will follow these steps:

### Step 1: Find the midpoint D of line segment AC.
The coordinates of point A are (-3, 1) and the coordinates of point C are (1, -2). The formula for finding the midpoint D(x, y) of a line segment connecting two points A(x1, y1) and C(x2, y2) is given by:

\[ D_x = \frac{x_1 + x_2}{2}, \quad D_y = \frac{y_1 + y_2}{2} \]

Substituting the coordinates of A and C into the formula:

\[ D_x = \frac{-3 + 1}{2} = \frac{-2}{2} = -1 \]
\[ D_y = \frac{1 + (-2)}{2} = \frac{-1}{2} = -\frac{1}{2} \]

Thus, the coordinates of point D are:

\[ D(-1, -\frac{1}{2}) \]

### Step 2: Use the coordinates of points B and D to find the equation of line BD.
The coordinates of point B are (4, 4) and point D are (-1, -\frac{1}{2}). The slope (m) of line BD can be calculated using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the coordinates of B and D:

\[ m = \frac{-\frac{1}{2} - 4}{-1 - 4} = \frac{-\frac{1}{2} - \frac{8}{2}}{-5} = \frac{-\frac{9}{2}}{-5} = \frac{9}{10} \]

### Step 3: Write the equation of the line in point-slope form.
The point-slope form of the equation of a line is given by:

\[ y - y_1 = m(x - x_1) \]

Using point B(4, 4) and the slope we calculated:

\[ y - 4 = \frac{9}{10}(x - 4) \]

### Step 4: Simplify the equation.
Distributing the slope on the right side:

\[ y - 4 = \frac{9}{10}x - \frac{36}{10} \]

Adding 4 (or \frac{40}{10}) to both sides:

\[ y = \frac{9}{10}x - \frac{36}{10} + \frac{40}{10} \]
\[ y = \frac{9}{10}x + \frac{4}{10} \]
\[ y = \frac{9}{10}x + \frac{2}{5} \]

### Final Equation
Thus, the equation of the median BD is:

\[ y = \frac{9}{10}x + \frac{2}{5} \]

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