Find the third verted of a triangle whose centroid is origin and two vertices are (2,4,6) and (-2,-2,1)

Answer

To find the third vertex of a triangle whose centroid is at the origin (0, 0, 0) and whose two vertices are given as A(2, 4, 6) and B(-2, -2, 1), we can follow these steps:

### Step 1: Understand the formula for the centroid
The centroid (G) of a triangle with vertices A(x1, y1, z1), B(x2, y2, z2), and C(x3, y3, z3) is given by the formula:
\[ G = \left( \frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3}, \frac{z1 + z2 + z3}{3} \right) \]
Since the centroid is at the origin, we have:
\[ G = (0, 0, 0) \]

### Step 2: Set up equations based on the centroid formula
Using the coordinates of points A and B, we can set up the following equations for the x, y, and z coordinates of the centroid:

1. For the x-coordinate:
\[ \frac{2 + (-2) + x}{3} = 0 \]
2. For the y-coordinate:
\[ \frac{4 + (-2) + y}{3} = 0 \]
3. For the z-coordinate:
\[ \frac{6 + 1 + z}{3} = 0 \]

### Step 3: Solve for the x-coordinate
From the first equation:
\[ \frac{2 - 2 + x}{3} = 0 \implies \frac{x}{3} = 0 \implies x = 0 \]

### Step 4: Solve for the y-coordinate
From the second equation:
\[ \frac{4 - 2 + y}{3} = 0 \implies \frac{2 + y}{3} = 0 \implies 2 + y = 0 \implies y = -2 \]

### Step 5: Solve for the z-coordinate
From the third equation:
\[ \frac{6 + 1 + z}{3} = 0 \implies \frac{7 + z}{3} = 0 \implies 7 + z = 0 \implies z = -7 \]

### Step 6: Write the coordinates of the third vertex
Thus, the coordinates of the third vertex C are:
\[ C(0, -2, -7) \]

### Final Answer
The third vertex of the triangle is \( C(0, -2, -7) \).

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