A (1, -5), B (2, 2) and C (-2, 4) are the vertices of triangle ABC. find the equation of:
the median of the triangle through A. 

Answer

To find the equation of the median of triangle ABC through vertex A, we will follow these steps:

### Step 1: Identify the vertices of the triangle
The vertices of triangle ABC are given as:
- A(1, -5)
- B(2, 2)
- C(-2, 4)

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

Calculating the coordinates of D:
\[ D\left(\frac{2 + (-2)}{2}, \frac{2 + 4}{2}\right) = D\left(\frac{0}{2}, \frac{6}{2}\right) = D(0, 3) \]

### Step 3: Calculate the slope of line AD
Now, we need to find the slope of the line AD, where A is (1, -5) and D is (0, 3). The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of A and D:
\[ m = \frac{3 - (-5)}{0 - 1} = \frac{3 + 5}{-1} = \frac{8}{-1} = -8 \]

### Step 4: Use point-slope form to find the equation of line AD
We can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
Using point A(1, -5) and the slope m = -8:
\[ y - (-5) = -8(x - 1) \]
Simplifying this equation:
\[ y + 5 = -8x + 8 \]
\[ y = -8x + 8 - 5 \]
\[ y = -8x + 3 \]

### Step 5: Rearranging to standard form
To write the equation in standard form:
\[ 8x + y - 3 = 0 \]

Thus, the equation of the median through vertex A is:
\[ 8x + y - 3 = 0 \]