Question

If the coordinates of vertices of a triangle is always rational then the triangle cannot be:​

Answer 1

Explanation:

If the coordinates of vertices of a triangle is always rational then the triangle cannot be (1) Scalene (2) Isosceles (3) Rightangle (4) Equilateral. ... Thus, there is a contradiction as a Rational Number equals an Irrational Number. Hence, coordinates of vertices of an equilateral triangle cannot be all rational.

Without loss of generality we can chose the co-ordinates ( we can make this by proper shift of co-ordinates) as A= (0,0), B= (x,y), C=(a,b)

Now slope of AB = y/x is rationalSlope of AC = b/a is rational

So tan (BAC) = ((y/x) – (a/b))/ ( 1 + ay/(bx)) which is rationsl

As tan 60 = sqrt(3) there cannot be any point with rational coefficient so angle BAC cannot be 60 degreeAs we cannot find any rational point on the kline at 60 degrees so getting an equilateral triangle is not possible.  

Hence proved

Hope this helps mate ☺️.