Question

# If two vertices of an equilateral triangle have integral coordinates, then the coordinates of the third vertex are both irrational both are integersboth are rationalat least one is irrational

Solution

## The correct option is D at least one is irrationalLet 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(y2−y3)+x2(y3−y1)+x3(y1−y2)] 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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