The centroid of a triangle formed by the lines $x=0, y=0$ and $x+y-1=0$ is:

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Accepted Answer

To determine the centroid of the triangle formed by the lines $x=0$, $y=0$, and $x+y-1=0$, we first need to identify the vertices of the triangle.

1. **Vertices of the Triangle:**
   - The line $x=0$ is the y-axis.
   - The line $y=0$ is the x-axis.
   - The line $x+y-1=0$ can be rewritten as $y=1-x$.

The vertices of the triangle are the points of intersection of these lines:
   - Intersection of $x=0$ and $y=0$ is the origin: $(0,0)$.
   - Intersection of $x=0$ and $x+y-1=0$: Substituting $x=0$ into $x+y-1=0$ gives $y=1$. So, the point is $(0,1)$.
   - Intersection of $y=0$ and $x+y-1=0$: Substituting $y=0$ into $x+y-1=0$ gives $x=1$. So, the point is $(1,0)$.

Therefore, the vertices of the triangle are $(0,0)$, $(0,1)$, and $(1,0)$.

2. **Calculating the Centroid:**
   The centroid $(x_c, y_c)$ of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by the formula:
   $$
   x_c = \frac{x_1 + x_2 + x_3}{3}, \quad y_c = \frac{y_1 + y_2 + y_3}{3}
   $$

Substituting the coordinates of the vertices:
   $$
   x_c = \frac{0 + 0 + 1}{3} = \frac{1}{3}
   $$
   $$
   y_c = \frac{0 + 1 + 0}{3} = \frac{1}{3}
   $$

Thus, the centroid of the triangle is $\left(\frac{1}{3}, \frac{1}{3}\right)$.

3. **Justification for Option 3:**
   The calculated centroid $\left(\frac{1}{3}, \frac{1}{3}\right)$ matches exactly with Option 3. Therefore, Option 3 is the correct answer.

4. **Explanation of Incorrect Options:**
   - **Option 1: $(\frac{2}{3}, \frac{2}{3})$**: This option is incorrect because the calculated centroid coordinates are $\left(\frac{1}{3}, \frac{1}{3}\right)$, not $\left(\frac{2}{3}, \frac{2}{3}\right)$.
   - **Option 2: $(0, \frac{2}{3})$**: This option is incorrect because the x-coordinate of the centroid is $\frac{1}{3}$, not 0.
   - **Option 4: $(0, \frac{1}{3})$**: This option is incorrect because the x-coordinate of the centroid is $\frac{1}{3}$, not 0.

In conclusion, the correct answer is Option 3, $\left(\frac{1}{3}, \frac{1}{3}\right)$, as it accurately represents the centroid of the triangle formed by the given lines.