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

Options

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

Answer

C

Explanation

Let `A(x_(1),y_(1)),B (x_(2),y_(2))` and `Cx_(3),y_(3))` be the verticales of an equilateral 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 " Delta ABC`
`rArr Delta = (1)/(2)|{:(,x_(1),y_(1),1),(,x_(2),y_(2),1),(,x_(3),y_(3),1):}|="A rationa number"`
But, `Delta = (sqrt(3))/(4)("Side")^(2)=(sqrt(3))/(4) xx" A rational number"`
`rArr " Delta = "An irrational number"`
Thus , we are arrive at contradiction. Therefore, our supposition is worng .
Hence, at lest one coordinate of C is irrational.