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

Options

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

Answer

C

Explanation

Let `A(x_(1),x_(2))B(x_(2),y_(2)) and C(x_(3),y_(3))` be the vertice of an equlateral triangle ABC such that `x_(1),x_(2) and y_(1),y_(2)` are integers.
If we assume that none of the coordinates of the vertex C are irrational, then we find that
`Delta`= Area of `DeltaABC`
`rArr Delta=(1)/(2)|{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=A` rational number
But, `Delta=(sqrt(3))/(4) ("Side")^(2)=(sqrt(3))/(4)xxA` rational number
`rArr Delta`= an irrational number
Thus, we arrive at a conradinte. Therefore , our supposition is wrong
Hence, at least at least one coordinate of C is irrational.