If two vertices of an equilaterla triangle have integral coordinates, then the third vetex will have

Options

A. Integral coordinates
B. Coordinates which are not rational
C. Coordinates which are rational
D. Nothing can be said

Answer

To solve the problem, we need to determine the nature of the coordinates of the third vertex of an equilateral triangle when the first two vertices have integral coordinates.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two vertices of an equilateral triangle with integral coordinates, say \( A(x_1, y_1) \) and \( B(x_2, y_2) \). We need to find the nature of the coordinates of the third vertex \( C(x_3, y_3) \).

2. **Area of the Triangle**:
The area \( A \) of triangle \( ABC \) can be calculated using the determinant formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
This area must also equal the area calculated using the formula for an equilateral triangle:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
where \( s \) is the length of a side of the triangle.

3. **Finding the Length of the Side**:
The length of the side \( s \) can be calculated as the distance between points \( A \) and \( B \):
\[ s = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Therefore, \( s^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 \).

4. **Area in Terms of Side Length**:
Substituting \( s^2 \) into the area formula for the equilateral triangle gives:
\[ A = \frac{\sqrt{3}}{4} \left((x_2 - x_1)^2 + (y_2 - y_1)^2\right) \]

5. **Nature of the Area**:
Since \( (x_2 - x_1)^2 + (y_2 - y_1)^2 \) is rational (as it is the sum of squares of integers), the area \( A \) becomes:
\[ A = \frac{\sqrt{3}}{4} \cdot \text{(rational)} \]
Since \( \sqrt{3} \) is irrational, \( A \) itself is irrational.

6. **Conclusion about the Third Vertex**:
For the area calculated using the determinant to be equal to the area calculated using the side length, the coordinates of the third vertex \( C(x_3, y_3) \) must be such that the overall area remains irrational. This means that at least one of the coordinates \( x_3 \) or \( y_3 \) must be irrational.

### Final Answer:
Thus, if two vertices of an equilateral triangle have integral coordinates, then the third vertex will have coordinates that are not both rational, implying that at least one of them is irrational.