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

detailed solution

Correct option is C

Let Ax1,y1,Bx2,y2 and  Cx3,y3  be the vertices of an equilateral triangle ABC such that x1,x2 and y1,y2 are integers. If we assume that none of the coordinates of the vertex C are irrational, then we find thatΔ= Area of ΔABC⇒ Δ=12x1    y1    1x2    y2    1x3    y3    1=A rational numberBut, Δ=34( Side )2=34×A rational number⇒ Δ= an irrational number.Thus, we arrive at a contradiction. Therefore, our supposition is wrong. Hence, at least one coordinate of C is irrational