Given the vertices A(10, 4), B(-4, 9) and C(-2, -1) of `DeltaABC`, find
the equation of the median through A

Answer

To find the equation of the median through vertex A of triangle ABC with vertices A(10, 4), B(-4, 9), and C(-2, -1), we will follow these steps:

### Step 1: Find the midpoint D of segment BC
To find the midpoint D of the line segment BC, we use the midpoint formula:
\[ D\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
where \(B(-4, 9)\) and \(C(-2, -1)\).

Calculating the coordinates of D:
\[ D\left(\frac{-4 + (-2)}{2}, \frac{9 + (-1)}{2}\right) = D\left(\frac{-6}{2}, \frac{8}{2}\right) = D(-3, 4) \]

### Step 2: Determine the slope of line AD
Now, we need to find the slope of the line AD, where A(10, 4) and D(-3, 4). The slope \(m\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of A and D:
\[ m = \frac{4 - 4}{-3 - 10} = \frac{0}{-13} = 0 \]

### Step 3: Write the equation of line AD
Since the slope \(m = 0\), the line AD is horizontal. The equation of a horizontal line passing through a point (x, y) is:
\[ y = k \]
where \(k\) is the y-coordinate of the point. Since point A has a y-coordinate of 4, the equation of line AD is:
\[ y = 4 \]

### Final Answer
The equation of the median through vertex A is:
\[ \boxed{y = 4} \]