If the coordinates of vertices of a triangle is always rational then the triangle cannot be

Options

A. Scalene
B. Isosceles
C. Rightangle
D. Equilateral

Answer

To solve the problem, we need to determine what type of triangle cannot be formed if all the vertices of the triangle have rational coordinates.

### Step-by-step Solution:

1. **Understanding Rational Coordinates**:
- The vertices of the triangle are given as \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \), where all \( x_i \) and \( y_i \) are rational numbers.

2. **Area of Triangle Formula**:
- The area \( A \) of a triangle with vertices at \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
- Since all \( x_i \) and \( y_i \) are rational, the area \( A \) will also be a rational number.

3. **Types of Triangles**:
- The triangle can be scalene, isosceles, or right-angled, and for these types, the area can be calculated using rational base and height, which will yield a rational area.

4. **Equilateral Triangle Area**:
- For an equilateral triangle with side length \( a \), the area is given by:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
- If \( a \) is rational, then \( a^2 \) is rational. However, \( \sqrt{3} \) is an irrational number.

5. **Conclusion**:
- Therefore, the area of an equilateral triangle, which is \( \frac{\sqrt{3}}{4} a^2 \), will be irrational when \( a \) is rational. This means that if the vertices of a triangle are all rational, the triangle cannot be equilateral.

### Final Answer:
The triangle cannot be an **equilateral triangle** if all its vertices have rational coordinates.