Find the lengths of the medians of the triangle whose vertices are `A (2,-3, 1), B (-6,5,3), C(8,7,- 7).`

Answer

To find the lengths of the medians of the triangle with vertices \( A(2, -3, 1) \), \( B(-6, 5, 3) \), and \( C(8, 7, -7) \), we will follow these steps:

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

1. **Midpoint \( M \) of side \( BC \)**:
\[ M = \left( \frac{x_B + x_C}{2}, \frac{y_B + y_C}{2}, \frac{z_B + z_C}{2} \right) \]
\[ M = \left( \frac{-6 + 8}{2}, \frac{5 + 7}{2}, \frac{3 + (-7)}{2} \right) = \left( \frac{2}{2}, \frac{12}{2}, \frac{-4}{2} \right) = (1, 6, -2) \]

2. **Midpoint \( N \) of side \( AC \)**:
\[ N = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2}, \frac{z_A + z_C}{2} \right) \]
\[ N = \left( \frac{2 + 8}{2}, \frac{-3 + 7}{2}, \frac{1 + (-7)}{2} \right) = \left( \frac{10}{2}, \frac{4}{2}, \frac{-6}{2} \right) = (5, 2, -3) \]

3. **Midpoint \( O \) of side \( AB \)**:
\[ O = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}, \frac{z_A + z_B}{2} \right) \]
\[ O = \left( \frac{2 - 6}{2}, \frac{-3 + 5}{2}, \frac{1 + 3}{2} \right) = \left( \frac{-4}{2}, \frac{2}{2}, \frac{4}{2} \right) = (-2, 1, 2) \]

### Step 2: Calculate the lengths of the medians.

1. **Length of median \( AM \)**:
\[ AM = \sqrt{(x_M - x_A)^2 + (y_M - y_A)^2 + (z_M - z_A)^2} \]
\[ AM = \sqrt{(1 - 2)^2 + (6 - (-3))^2 + (-2 - 1)^2} \]
\[ = \sqrt{(-1)^2 + (6 + 3)^2 + (-3)^2} = \sqrt{1 + 81 + 9} = \sqrt{91} \]

2. **Length of median \( BN \)**:
\[ BN = \sqrt{(x_N - x_B)^2 + (y_N - y_B)^2 + (z_N - z_B)^2} \]
\[ BN = \sqrt{(5 - (-6))^2 + (2 - 5)^2 + (-3 - 3)^2} \]
\[ = \sqrt{(5 + 6)^2 + (-3)^2 + (-6)^2} = \sqrt{11^2 + 9 + 36} = \sqrt{121 + 9 + 36} = \sqrt{166} \]

3. **Length of median \( CO \)**:
\[ CO = \sqrt{(x_O - x_C)^2 + (y_O - y_C)^2 + (z_O - z_C)^2} \]
\[ CO = \sqrt{(-2 - 8)^2 + (1 - 7)^2 + (2 - (-7))^2} \]
\[ = \sqrt{(-10)^2 + (-6)^2 + (2 + 7)^2} = \sqrt{100 + 36 + 81} = \sqrt{217} \]

### Final Results:
- Length of median \( AM = \sqrt{91} \)
- Length of median \( BN = \sqrt{166} \)
- Length of median \( CO = \sqrt{217} \)