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  • . State whether the statements are true or false. If the vertices of a triangle have integral coordinates, then the triangle can’t be equilateral.

State whether the statements are true or false.

If the vertices of a triangle have integral coordinates, then the triangle can’t be equilateral.

Answers (1)

Let ABC be a triangle with vertices A(x1,y1), B (x2,y2) and C (x3, y3) where xi, yi, i=1,2,3 are integers    

Then area of ABC=\frac{1}{2}\left [ x_{1}\left ( y_{2}-y_{3} \right ) +x_{2}\left ( y_{3}-y_{1} \right ) +x_{3}\left ( y_{1}-y_{2} \right ) \right ]

 Since, xi and yi all are integers but  \frac{1}{2} is a rational number.

 So, the result comes out to be a rational number. i.e . Area of ABC=a rational number 

Suppose, ABC be an equilateral triangle, then area of ABC is=\frac{\sqrt{3}}{4}\left ( AB \right )^{2} 

It is given that vertices are integral coordinates, it means the value of coordinates is in whole

 number. Therefore, the value of (AB)2 is also an integer. 

\frac{\sqrt{3}}{4} (positive integer)   

 But, \sqrt{3} is an irrational number 

  Area of triangle ABC=an ir-rational number  

This contradicts the fact that the area is a rational number

  Hence, the given statement is true.

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