A (5, 4), B (-3, -2) and C (1, -8) are the vertices of a triangle ABC. Find :
the slope of the median AD .

Answer

To find the slope of the median AD in triangle ABC with vertices A(5, 4), B(-3, -2), and C(1, -8), we will follow these steps:

### Step 1: Find the coordinates of point D
Point D is the midpoint of line segment BC. The formula for the midpoint D(x, y) of two points B(x1, y1) and C(x2, y2) is given by:

\[ D = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Here, the coordinates of B are (-3, -2) and C are (1, -8).

Using the midpoint formula:

\[ D = \left( \frac{-3 + 1}{2}, \frac{-2 + (-8)}{2} \right) \]

Calculating the x-coordinate:

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

Calculating the y-coordinate:

\[ D_y = \frac{-2 - 8}{2} = \frac{-10}{2} = -5 \]

Thus, the coordinates of D are:

\[ D(-1, -5) \]

### Step 2: Use the slope formula to find the slope of AD
The slope \( m \) of a line passing through two points A(x1, y1) and D(x2, y2) is given by the formula:

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

Here, the coordinates of A are (5, 4) and D are (-1, -5). We can assign:

- \( x_1 = 5 \), \( y_1 = 4 \) (for point A)
- \( x_2 = -1 \), \( y_2 = -5 \) (for point D)

Substituting these values into the slope formula:

\[ m = \frac{-5 - 4}{-1 - 5} \]

Calculating the numerator:

\[ -5 - 4 = -9 \]

Calculating the denominator:

\[ -1 - 5 = -6 \]

Thus, we have:

\[ m = \frac{-9}{-6} = \frac{9}{6} = \frac{3}{2} \]

### Final Answer
The slope of the median AD is:

\[ \frac{3}{2} \]

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