Find the third vertex of triangle whose centroid is origin and two vertices are (1,2,3) and (0,-2,-5)

Answer

To find the third vertex of the triangle whose centroid is at the origin and two vertices are given as (1, 2, 3) and (0, -2, -5), 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:
\[ G = \left( \frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3}, \frac{z1 + z2 + z3}{3} \right) \]

### Step 2: Set up the equations
Given that the centroid is at the origin (0, 0, 0), we can set up the following equations based on the coordinates of the vertices:
1. For the x-coordinates:
\[ \frac{1 + 0 + x3}{3} = 0 \]
2. For the y-coordinates:
\[ \frac{2 + (-2) + y3}{3} = 0 \]
3. For the z-coordinates:
\[ \frac{3 + (-5) + z3}{3} = 0 \]

### Step 3: Solve for x3
From the first equation:
\[ \frac{1 + 0 + x3}{3} = 0 \implies 1 + x3 = 0 \implies x3 = -1 \]

### Step 4: Solve for y3
From the second equation:
\[ \frac{2 - 2 + y3}{3} = 0 \implies 0 + y3 = 0 \implies y3 = 0 \]

### Step 5: Solve for z3
From the third equation:
\[ \frac{3 - 5 + z3}{3} = 0 \implies -2 + z3 = 0 \implies z3 = 2 \]

### Step 6: Write the coordinates of the third vertex
Now that we have found the values of x3, y3, and z3, the coordinates of the third vertex C are:
\[ C(-1, 0, 2) \]

### Final Answer
The third vertex of the triangle is \((-1, 0, 2)\).

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