if A(-2,3),B(a,0),C(4,b) and D(1,2) are the vertices of a parallelogram ABCD find the values of a and b hence find the lengths of its sides
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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...
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...
Answered by
1
Answer:
a = 1, b = 1, AB = √10, BC = √10
Step-by-step explanation:
As we know the diagonals of a parallelogram bisect each other, we can take their midpoints.
Midpoint of diagonal AC = Midpoint of diagonal BD.
Given A(-2,1),B(a,0),C(4,b),D(1,2).
By using the formula (x1 + x2)/2 and (y1 + y2)/2 we have
(-2 + 4)/2, (1 + b)/2 = (a + 1)/2 , (0 + 2)/2
on simplifying we get (b+1)/2 = 1 and (a+1)/2 = 1
So a = 1 and b = 1
Using the distance formula the length of parallelogram can be obtained.
AB = √(x2 - x1)^2 + (y2 - y1)^2
= √(1 -(- 2)^2 + (0 -1)^2
= √9 -1
AB = √10
BC = √(4 - 1)^2 + (1 - 0)^2
BC = √10
It is a rhombus since the sides are equal.
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