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Question

If two vertices of an equilateral triangle have integral coordinates, then the coordinates of the third vertex are 
  1. both irrational 
  2. both are integers
  3. both are rational
  4. at least one is irrational


Solution

The correct option is D at least one is irrational
Let the vertices of the equilateral triangle be (x1,y1),(x2,y2) and (x3,y3)

Area of an equilateral triangle =34(side)2=irrational

Also, the area of triangle can be written as
Δ=12[x1(y2y3)+x2(y3y1)+x3(y1y2)]
If all the vertices of the triangle are rational, then Δ=rational

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

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