A triangle ABC is given by A(2, 5), B(-1, -1), C (3, 1). The equation of median drawn on BC from A, is :

Options

A. 2 X + Y - 9 = 0
B. 5X - Y - 5 = 0
C. 3 X + Y = - 9
D. None of these

Answer

To find the equation of the median drawn from vertex A to side BC of triangle ABC with vertices A(2, 5), B(-1, -1), and C(3, 1), we will follow these steps:

### Step 1: Find the midpoint D of side BC
The coordinates of points B and C are B(-1, -1) and C(3, 1). We can find the midpoint D using the midpoint formula:

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

Substituting the coordinates of B and C:

\[ D\left(\frac{-1 + 3}{2}, \frac{-1 + 1}{2}\right) = D\left(\frac{2}{2}, \frac{0}{2}\right) = D(1, 0) \]

### Step 2: Find the slope of line AD
Now, we need to find the slope of the line AD, where A(2, 5) and D(1, 0). The slope (m) is given by:

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

Substituting the coordinates of A and D:

\[ m = \frac{0 - 5}{1 - 2} = \frac{-5}{-1} = 5 \]

### Step 3: 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 given by:

\[ y - y_1 = m(x - x_1) \]

Using point A(2, 5) and the slope we found:

\[ y - 5 = 5(x - 2) \]

### Step 4: Simplify the equation
Now, we will simplify this equation:

\[ y - 5 = 5x - 10 \]

Adding 5 to both sides:

\[ y = 5x - 5 \]

### Step 5: Write the equation in standard form
To write the equation in standard form (Ax + By + C = 0), we rearrange it:

\[ 5x - y - 5 = 0 \]

Thus, the equation of the median drawn from A to BC is:

\[ 5x - y - 5 = 0 \]