In `DeltaABC`, A (3, 5), B (7, 8) and C(1, -10). Find the equation of the median through A.

Answer

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

### Step 1: Identify the coordinates of points A, B, and C.
- A = (3, 5)
- B = (7, 8)
- C = (1, -10)

### Step 2: Find the midpoint D of segment BC.
The midpoint D can be calculated using the midpoint formula:
\[ D = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Where \( (x_1, y_1) \) are the coordinates of B and \( (x_2, y_2) \) are the coordinates of C.

Substituting the coordinates:
\[ D = \left( \frac{7 + 1}{2}, \frac{8 + (-10)}{2} \right) = \left( \frac{8}{2}, \frac{-2}{2} \right) = (4, -1) \]

### Step 3: Find the slope of line AD.
The slope \( m \) of line AD can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Where \( (x_1, y_1) \) are the coordinates of A and \( (x_2, y_2) \) are the coordinates of D.

Substituting the coordinates:
\[ m = \frac{-1 - 5}{4 - 3} = \frac{-6}{1} = -6 \]

### Step 4: Use the point-slope form to find the equation of line AD.
The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Using point A (3, 5) and the slope \( m = -6 \):
\[ y - 5 = -6(x - 3) \]

### Step 5: Simplify the equation.
Expanding the equation:
\[ y - 5 = -6x + 18 \]
Rearranging gives:
\[ 6x + y - 23 = 0 \]

### Final Result:
The equation of the median through point A is:
\[ 6x + y - 23 = 0 \]

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