Question

if a (-2,1) , b(9,0) , c(4,b) and d (1,a) are vertices of a parallelogram ,abcd . find the values of a and b and hence find the length of each side

Answer 1

here you wrote wrong question, actually every parallelogram have a basic property that diagonals of it bisect each other . But here we see
(9 + 1)/2 ≠ ( -2 + 4)/2 [ x - co- ordinate of midpoint of diagonals doesn't match ]

Your correct question is ----> If A (-2,1) , B(a,0) , C(4,b) and D(1,2) are vertices of a parallelogram ,abcd . find the values of a and b and hence find the length of each side ?

solution :- we know, diagonals of parallelogram bisect each other .
Use midpoint section formula ,
e.g., x = (x₁ + x₂)/2 , y = (y₁ + y₂) , use this here
Midpoint of AC = midpoint of BD [ according to property of ||gm ]

midpoint of AC = [(-2 + 4)/2 , (1 + b)/2 ]
Midpoint of BD = [(a + 1)/2 , (0 + 2)/2 ]

x - co-ordinate of midpoint of AC = x - co-ordinate of midpoint of BD
(-2+4)/2 = (a +1)/2
2 = a + 1 ⇒ a = 1

Similarly , y - co- ordinate of midpoint of AC = y - co-ordinate of midpoint of BD
(1 + b)/2 = (0+2)/2
1 + b = 2 ⇒b = 1

Now, length of AB = distance between (-2,1) and (1,0)
= √{(-2-1)² + (1)²} = √10 unit
Length of BC = distance between (1,0) and (4,1)
= √{(4-1)² + (1-0)²} = √10 unit

Hence, AB = BC = CD = DA = √10 unit