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 :
at least one coordinate irrational
Solution :
If none of the coordinates are irrational then
\[\Delta = \frac{1}{2}\left| \begin{array}{l}
{x_1}\,\,\,\,\,{y_1}\,\,\,\,\,1\\
{x_2}\,\,\,\,\,{y_2}\,\,\,\,\,1\\
{x_3}\,\,\,\,\,{y_3}\,\,\,\,\,1
\end{array} \right| = \frac{1}{2} \times {\rm{rational}} = {\rm{rational}}\]
But the area of an equilateral triangle
$$ = \frac{{\sqrt 3 }}{4} \times {\left( {{\text{side}}} \right)^2} = \frac{{\sqrt 3 }}{4} \times {\text{rational}} = {\text{irrational}}$$
So, both the coordinates of the third vertex cannot be rational.