The equation(s) of the medians of the triangle formed by the points (4, 8), (3, 2) and 5, -6) is/are :

Options

A. `x=4`
B. `x=5y-3`
C. `2x+3y-12=0`
D. `22x+3y-92=0`

Answer

To find the equations of the medians of the triangle formed by the points \( A(4, 8) \), \( B(3, 2) \), and \( C(5, -6) \), we will follow these steps:

### Step 1: Find the midpoints of the sides of the triangle.

1. **Midpoint D of BC**:
\[ D = \left( \frac{x_B + x_C}{2}, \frac{y_B + y_C}{2} \right) = \left( \frac{3 + 5}{2}, \frac{2 + (-6)}{2} \right) = \left( \frac{8}{2}, \frac{-4}{2} \right) = (4, -2) \]

2. **Midpoint E of AC**:
\[ E = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) = \left( \frac{4 + 5}{2}, \frac{8 + (-6)}{2} \right) = \left( \frac{9}{2}, \frac{2}{2} \right) = \left( \frac{9}{2}, 1 \right) \]

3. **Midpoint F of AB**:
\[ F = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) = \left( \frac{4 + 3}{2}, \frac{8 + 2}{2} \right) = \left( \frac{7}{2}, 5 \right) \]

### Step 2: Find the equations of the medians.

1. **Median AD**:
- Points: \( A(4, 8) \) and \( D(4, -2) \)
- Slope of AD:
\[ m_{AD} = \frac{y_D - y_A}{x_D - x_A} = \frac{-2 - 8}{4 - 4} = \frac{-10}{0} \quad \text{(undefined, vertical line)} \]
- Equation of median AD:
\[ x = 4 \]

2. **Median BE**:
- Points: \( B(3, 2) \) and \( E\left( \frac{9}{2}, 1 \right) \)
- Slope of BE:
\[ m_{BE} = \frac{y_E - y_B}{x_E - x_B} = \frac{1 - 2}{\frac{9}{2} - 3} = \frac{-1}{\frac{3}{2}} = -\frac{2}{3} \]
- Equation of median BE:
\[ y - y_B = m_{BE}(x - x_B) \implies y - 2 = -\frac{2}{3}(x - 3) \]
\[ y - 2 = -\frac{2}{3}x + 2 \implies y = -\frac{2}{3}x + 4 \]
- Rearranging gives:
\[ 2x + 3y - 12 = 0 \]

3. **Median CF**:
- Points: \( C(5, -6) \) and \( F\left( \frac{7}{2}, 5 \right) \)
- Slope of CF:
\[ m_{CF} = \frac{y_F - y_C}{x_F - x_C} = \frac{5 - (-6)}{\frac{7}{2} - 5} = \frac{11}{\frac{-3}{2}} = -\frac{22}{3} \]
- Equation of median CF:
\[ y - y_C = m_{CF}(x - x_C) \implies y + 6 = -\frac{22}{3}(x - 5) \]
\[ y + 6 = -\frac{22}{3}x + \frac{110}{3} \]
\[ 3y + 18 = -22x + 110 \implies 22x + 3y - 92 = 0 \]

### Final Equations of the Medians:
1. Median AD: \( x = 4 \)
2. Median BE: \( 2x + 3y - 12 = 0 \)
3. Median CF: \( 22x + 3y - 92 = 0 \)