Question

if A(-2,1) B(a,0) C(4,b) D(1,3) are the vertices of a parallelogram then find a and b and hence find length of its sides

Answer 1

Given :
A(-2,1), B(a,0), C(4,b) , D(1,2) are the four vertices of a parallelogram.

We know that the diagonals of a parallelogram bisect each other.

Coordinates of midpoint of AC = coordinates of midpoint BD

For midpoint : A(-2,1) , C(4,b)
x1= -2, y1=1,x2=4,y2= b
=[(-2+4)/2 ,(1+b)/2) ]= [(2/2,(1+b)/2)] = [1,(1+b)/2]
[coordinates of midpoint of a line segment joining the points(x1,y1) & (x2,y2) = (x1+x2/2, y1+y2/2)]
For midpoint BD : B(a,0) , D(1,2)
x1= a, y1=0,x2=1, y2= 2
=[(a+1)/2 ,0+2/2) = [(a+1)/2,2/2) = [(a+1)/2,1)
Coordinates of midpoint of AC = coordinates of midpoint BD
[1,(1+b)/2] =[ (a+1)/2),1)]
On equating the coordinates ,
(a+1)/2 = 1
a+1 = 2×1
a+1 = 2
a = 2-1
a= 1
(1+b)/2 = 1
1+b = 2×1
1+b = 2
b = 2-1
b = 1
Hence, the coordinates of A (-2,1) , B(1,0), C(4,1), D(1,2)
In ||gm ABCD ,
AB = CD & BC = AD
[Opposite sides of a ||gm are equal]
Length of AB = √(x1-x2)² + (y1-y2)²
[By distance formula]
Length of AB = √(-2-1)² +(1-0)² = √(-3)² + 1²
Length of AB = √9 + 1 =√10
Length of AB = CD = √10
Length of BC = √(x1-x2)² + (y1-y2)²
Length of BC = √(1- 4)² + (0+1)² =√3² + 1²
Length of BC = √9+1= √10
Length of BC = AD = √10
Hence, the values of a = 1 & b = 1 & the Length of the all the Sides of a ||gm are AB = CD = BC = AD = √10.

HOPE THIS WILL HELP YOU...