To find the centroid of the equilateral triangle ΔOAB where O is the origin (0,0) and A is a point on the x-axis, we can follow these steps:
### Step 1: Define the Points
Let the coordinates of point A on the x-axis be \( A(a, 0) \), where \( a \) is a rational number. The origin O is at \( O(0, 0) \). We need to find the coordinates of point B.
### Step 2: Determine the Coordinates of Point B
Since ΔOAB is an equilateral triangle, we can find the coordinates of point B using the properties of equilateral triangles. The length of OA is \( a \). The coordinates of point B can be derived using the rotation of point A around the origin by 60 degrees.
The coordinates of B after rotating A by 60 degrees can be calculated as:
- \( B\left(a \cos 60^\circ, a \sin 60^\circ\right) \)
- Since \( \cos 60^\circ = \frac{1}{2} \) and \( \sin 60^\circ = \frac{\sqrt{3}}{2} \), we have:
\[
B\left(a \cdot \frac{1}{2}, a \cdot \frac{\sqrt{3}}{2}\right) = \left(\frac{a}{2}, \frac{a\sqrt{3}}{2}\right)
\]
### Step 3: Calculate the Centroid
The centroid \( G \) of triangle ΔOAB is given by the formula:
\[
G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)
\]
Substituting the coordinates of points O, A, and B:
- \( O(0, 0) \)
- \( A(a, 0) \)
- \( B\left(\frac{a}{2}, \frac{a\sqrt{3}}{2}\right) \)
The coordinates of the centroid are:
\[
G\left(\frac{0 + a + \frac{a}{2}}{3}, \frac{0 + 0 + \frac{a\sqrt{3}}{2}}{3}\right)
\]
Simplifying this gives:
\[
G\left(\frac{a + \frac{a}{2}}{3}, \frac{\frac{a\sqrt{3}}{2}}{3}\right) = G\left(\frac{\frac{2a + a}{2}}{3}, \frac{a\sqrt{3}}{6}\right) = G\left(\frac{\frac{3a}{2}}{3}, \frac{a\sqrt{3}}{6}\right) = G\left(\frac{a}{2}, \frac{a\sqrt{3}}{6}\right)
\]
### Step 4: Analyze the Rationality of the Centroid
- The x-coordinate \( \frac{a}{2} \) is rational since \( a \) is rational.
- The y-coordinate \( \frac{a\sqrt{3}}{6} \) involves \( \sqrt{3} \), which is an irrational number. Therefore, \( \frac{a\sqrt{3}}{6} \) is irrational unless \( a = 0 \).
### Conclusion
Since the y-coordinate of the centroid is irrational for any rational \( a \neq 0 \), the centroid \( G \) is not a rational point. Thus, we conclude that the centroid of triangle ΔOAB is never a rational point.
