Question

If all 3 vertices of triangle have integral coordinates then that triangle cannot be equilateral

Answer 1

Answer:

If the vertices of a triangle have integral coordinates, then the triangle can't be equilateral. ... 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. But, √3 is an irrational number.

Step-by-step explanation:

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

Then, Area of ΔABC

Since, xi and yi all are integers but

is a rational number. So, the result comes out to be a rational number.

i.e. Area of ΔABC = a rational number

Suppose, ABC be an equilateral triangle, then Area of ΔABC is

[∵ AB = BC = CA]

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.

But, √3 is an irrational number.

⇒ Area of ΔABC = an ir-rational number

This is a contradiction to the fact that the area is a rational number.

Hence, the given statement is TRUE

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