A triangular plane ABC with centroid `(1,2,3)` cuts the coordinates axes at A, B, C, respectively
What are the intercepts made by the plane ABC on the axes?

Options

A. 3,6,9
B. 1,2,3
C. 1,4,9
D. 2,4,6

Answer

To find the intercepts made by the triangular plane ABC on the coordinate axes, we will follow these steps:

### Step 1: Understand the centroid of the triangle
The centroid of a triangle is given as \( (1, 2, 3) \). The coordinates of the centroid \( G \) of a triangle with vertices \( A(x_1, 0, 0) \), \( B(0, y_2, 0) \), and \( C(0, 0, z_3) \) can be calculated using the formula:
\[ G = \left( \frac{x_1 + 0 + 0}{3}, \frac{0 + y_2 + 0}{3}, \frac{0 + 0 + z_3}{3} \right) \]

### Step 2: Set up the equations for the centroid
From the formula for the centroid, we have:
\[ G = \left( \frac{x_1}{3}, \frac{y_2}{3}, \frac{z_3}{3} \right) \]
Given that \( G = (1, 2, 3) \), we can set up the following equations:
1. \( \frac{x_1}{3} = 1 \)
2. \( \frac{y_2}{3} = 2 \)
3. \( \frac{z_3}{3} = 3 \)

### Step 3: Solve for the intercepts
Now, we will solve each equation for \( x_1 \), \( y_2 \), and \( z_3 \):
1. From \( \frac{x_1}{3} = 1 \):
\[ x_1 = 3 \]
2. From \( \frac{y_2}{3} = 2 \):
\[ y_2 = 6 \]
3. From \( \frac{z_3}{3} = 3 \):
\[ z_3 = 9 \]

### Step 4: Write the intercepts
The intercepts made by the plane ABC on the axes are:
- \( A(x_1, 0, 0) = (3, 0, 0) \)
- \( B(0, y_2, 0) = (0, 6, 0) \)
- \( C(0, 0, z_3) = (0, 0, 9) \)

Thus, the intercepts are:
\[ \text{Intercepts: } (3, 6, 9) \]

### Final Answer
The intercepts made by the plane ABC on the axes are \( (3, 6, 9) \).

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