If the coordinates of the vertices of triangle `A B C` are `(-1,6),(-3,-9)` and `(5,-8)` , respectively, then find the equation of the median through `Cdot`

Explanation

To find the equation of the median through vertex C of triangle ABC with vertices A(-1, 6), B(-3, -9), and C(5, -8), we will follow these steps:

### Step 1: Find the midpoint of side AB
The midpoint \( P \) of line segment \( AB \) can be calculated using the midpoint formula:
\[ P\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
where \( A(-1, 6) \) and \( B(-3, -9) \).

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

### Step 2: Find the slope of line segment PC
Next, we need to find the slope of the line segment \( PC \). The slope \( m \) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
where \( P(-2, -\frac{3}{2}) \) and \( C(5, -8) \).

Calculating the slope:
\[ m = \frac{-8 - \left(-\frac{3}{2}\right)}{5 - (-2)} = \frac{-8 + \frac{3}{2}}{5 + 2} = \frac{-\frac{16}{2} + \frac{3}{2}}{7} = \frac{-\frac{13}{2}}{7} = -\frac{13}{14} \]

### Step 3: Write the equation of line PC
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) = (5, -8) \) and \( m = -\frac{13}{14} \).

Substituting the values:
\[ y - (-8) = -\frac{13}{14}(x - 5) \]
This simplifies to:
\[ y + 8 = -\frac{13}{14}x + \frac{65}{14} \]
Rearranging gives:
\[ y = -\frac{13}{14}x + \frac{65}{14} - 8 \]
Converting \( 8 \) into a fraction with a denominator of 14:
\[ y = -\frac{13}{14}x + \frac{65}{14} - \frac{112}{14} \]
This simplifies to:
\[ y = -\frac{13}{14}x - \frac{47}{14} \]

### Step 4: Convert to standard form
To convert this equation into standard form \( Ax + By + C = 0 \):
\[ \frac{13}{14}x + y + \frac{47}{14} = 0 \]
Multiplying through by 14 to eliminate the fraction:
\[ 13x + 14y + 47 = 0 \]

### Final Answer
The equation of the median through vertex C is:
\[ 13x + 14y + 47 = 0 \]