Find the point of intersection of the medians of the triangle with vertices (-1, -3, 4), (4, -2,-7), (2, 3, -8).

Answer

To find the point of intersection of the medians of the triangle with vertices A(-1, -3, 4), B(4, -2, -7), and C(2, 3, -8), we will calculate the centroid of the triangle. The centroid (G) is the point where the medians intersect, and its coordinates can be calculated using the formula:

\[ G\left(x, y, z\right) = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3}\right) \]

### Step-by-Step Solution:

1. **Identify the coordinates of the vertices:**
- A = (-1, -3, 4)
- B = (4, -2, -7)
- C = (2, 3, -8)

2. **Substitute the coordinates into the centroid formula:**
- \(x_1 = -1\), \(y_1 = -3\), \(z_1 = 4\)
- \(x_2 = 4\), \(y_2 = -2\), \(z_2 = -7\)
- \(x_3 = 2\), \(y_3 = 3\), \(z_3 = -8\)

3. **Calculate the x-coordinate of the centroid:**
\[ x = \frac{x_1 + x_2 + x_3}{3} = \frac{-1 + 4 + 2}{3} = \frac{5}{3} \]

4. **Calculate the y-coordinate of the centroid:**
\[ y = \frac{y_1 + y_2 + y_3}{3} = \frac{-3 - 2 + 3}{3} = \frac{-2}{3} \]

5. **Calculate the z-coordinate of the centroid:**
\[ z = \frac{z_1 + z_2 + z_3}{3} = \frac{4 - 7 - 8}{3} = \frac{-11}{3} \]

6. **Combine the coordinates to find the centroid:**
\[ G\left(x, y, z\right) = \left(\frac{5}{3}, \frac{-2}{3}, \frac{-11}{3}\right) \]

### Final Answer:
The point of intersection of the medians of the triangle is \(G\left(\frac{5}{3}, \frac{-2}{3}, \frac{-11}{3}\right)\).

---