Find the centroid of a triangle, mid points of whose sides are (1,2,-3), (3,0,1) and (-1,1,-4).

Options

A. `(1,-1,-2)`
B. `(1,1,-2)`
C. `(-1,1,-2)`
D. `(-1,-1,-2)`

Answer

To find the centroid of a triangle whose midpoints of the sides are given as \( P(1, 2, -3) \), \( Q(3, 0, 1) \), and \( R(-1, 1, -4) \), we can follow these steps:

### Step 1: Understand the formula for the centroid
The centroid \( G \) of a triangle with vertices at points \( (x_1, y_1, z_1) \), \( (x_2, y_2, z_2) \), and \( (x_3, y_3, z_3) \) is given by the formula:
\[ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right) \]

### Step 2: Substitute the coordinates of the midpoints
We will substitute the coordinates of the midpoints \( P(1, 2, -3) \), \( Q(3, 0, 1) \), and \( R(-1, 1, -4) \) into the centroid formula.

### Step 3: Calculate the x-coordinate of the centroid
\[ x_G = \frac{1 + 3 - 1}{3} = \frac{3}{3} = 1 \]

### Step 4: Calculate the y-coordinate of the centroid
\[ y_G = \frac{2 + 0 + 1}{3} = \frac{3}{3} = 1 \]

### Step 5: Calculate the z-coordinate of the centroid
\[ z_G = \frac{-3 + 1 - 4}{3} = \frac{-6}{3} = -2 \]

### Step 6: Combine the coordinates to find the centroid
Thus, the coordinates of the centroid \( G \) are:
\[ G(1, 1, -2) \]

### Final Answer:
The centroid of the triangle is \( G(1, 1, -2) \).

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