Question

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

• A. both irrational
• B. at least one is irrational
• C. both are rational
• D. both are integers

Answer 1

The correct option is B at least one is irrational

Let the vertices of the equilateral triangle be $(x_1,y_1),(x_2,y_2)$ and $(x_3,y_3)$

Area of an equilateral triangle $=\frac{\sqrt{3}}{4}{\text{(side)}}^2=\text{irrational}$

Also, the area of triangle can be written as
$\Delta =\frac{1}{2}\left[x_1\right(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2\left)\right]$
If all the vertices of the triangle are rational, then $\Delta =\text{rational}$

Both the statements are contradictory to each other. So at least one of the coordinates of the third vertex should be irrational.