To find the values of \( x \) and \( y \) such that the centroid of the triangle with vertices at \( (x, 0) \), \( (0, y) \), and \( (6, 3) \) is \( (3, 4) \), we can follow these steps:
### Step 1: Understand the formula for the centroid
The centroid \( G \) of a triangle with vertices at \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by the formula:
\[
G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)
\]
### Step 2: Substitute the given points into the centroid formula
For our triangle, the vertices are:
- \( (x, 0) \)
- \( (0, y) \)
- \( (6, 3) \)
Thus, the centroid \( G \) can be expressed as:
\[
G = \left( \frac{x + 0 + 6}{3}, \frac{0 + y + 3}{3} \right)
\]
### Step 3: Set the centroid equal to the given coordinates
We know that the centroid is \( (3, 4) \). Therefore, we can set up the following equations:
1. For the x-coordinate:
\[
\frac{x + 6}{3} = 3
\]
2. For the y-coordinate:
\[
\frac{y + 3}{3} = 4
\]
### Step 4: Solve the equations
**For the x-coordinate:**
Multiply both sides of the first equation by 3:
\[
x + 6 = 9
\]
Now, subtract 6 from both sides:
\[
x = 3
\]
**For the y-coordinate:**
Multiply both sides of the second equation by 3:
\[
y + 3 = 12
\]
Now, subtract 3 from both sides:
\[
y = 9
\]
### Step 5: Write the final answer
The values of \( (x, y) \) are:
\[
(x, y) = (3, 9)
\]
