Question

If two vertices of an equilateral triangle have integral coordinates, then the third vertex will have:

• A. integral coordinates
• B. coordinates which are rational
• C. at least one coordinate irrational
• D. coordinates which are irrational

Answer 1

The correct option is C at least one coordinate irrational

Let the vertices of the equilateral triangle be $(x_1,y_1),(x_2,y_2)$ and $(x_3,y_3)$
If none of $x_i$ and $y_i(i=1,2,3)$ are irrational, then
area of $\Delta =\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=$ rational
But the area of an equilateral triangle $=\frac{\sqrt{3}}{4}{\left(side\right)}^2=$ irrational
Thus, the two statements are contradictory,
Therefore, both the coordinates of the third vertex cannot be rational.