NDA Math Coordinate System and Straight Line

(c) Let vertex of
A=(0,0)
B=(2,2)∵△ABC is equilateral triangle.
∴‌‌AB=BC
⇒√(2−0)2+(2−0)2
‌‌=√(2−x)2+(2−y)2
⇒‌‌8=4+x2−4x+4+y2−4y
⇒‌‌x2+y2−4x−4y=0‌‌‌⋅⋅⋅⋅⋅⋅⋅(i)
and AB=AC
⇒‌‌√(2−0)2+(2−0)2
‌‌=√(x−0)2+(y−0)2
⇒‌‌8=x2+y2‌‌‌⋅⋅⋅⋅⋅⋅⋅(ii)
and AC=BC
⇒√x2+y2=√(x−2)2+(y−2)2
⇒‌‌x2+y2=x2+y2−4x−4y+8
⇒‌‌x+y=2‌‌‌⋅⋅⋅⋅⋅⋅⋅(iii)

From Eqs. (ii) and (iii)
8‌‌=x2+(2−x)2
x2+4+x2−4x‌‌=8
x2−2x−2‌‌=0
x‌‌=‌

2±√4+8

2

‌‌=‌

1±√3

2

Hence, third vertex atleast one irrational coordinate.
⇒ Area will also be irrational.