Question

Let $A(x_1,y_1),B(x_2,y_2),C(x_3,y_3)$ be three points. Area of triangle with vertices $A,B,C$ is given by $\frac{1}{2}\left|\Delta \right|$ where, $\Delta =\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$.If $△ABC$ is an equilateral triangle and $a=BC$ is a rational number, then $△$ must be

• A. an integer
• B. a rational number
• C. an irrational number
• D. an imaginary number

Answer 1

The correct option is C an irrational number

If $a$ is rational then $a^2$ is also rational
Now as $\Delta =\frac{\sqrt{3}}{4}{a}^2$
Then $\Delta$ is irrational