First slide

Cartesian plane

Question

If each of the vertices of a triangle has integer: coordinates, then the triangle will not be

Moderate

A

right angled
B

equilateral
C

isosceles
D

none of these

Solution

Let $A (x_{1}, y_{1}), B (x_{2}, y_{2})$ and $C (x_{3}, y_{3})$ be the
vertices of a $∆ A B C$ such that $x_{i}^{'} s$ and $y_{i}^{'} s$ are integers. Then
$Δ = Area of △ A B C = a$ rational number
But, if we assume that the triangle is equilateral, then
$Δ = \frac{\sqrt{3}}{4} (side)^{2} = \frac{\sqrt{3}}{4} \times an integer = a rational number$
So, the triangle cannot be equilateral

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