If A(-2,1), B(a,0), C(4,b), D(1,2) are the vertices of parallelogram ABCD. Find a and b and length of each sides
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Answer:
Step-by-step explanation:
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Given A(-2,1), B(a,0),C(4,b),
and D(1,2) are vertices of a
Parallelogram ABCD.
We know that the diagonals
of a parallelogram bisects each
other .
So the midpoint of the diagonals
AC and DB should be same.
Now , we find the midpoint of
AC and DB by using
[ (x1+x2)/2 , (y1+y2)/2 ] formula.
Midpoint of AC=[(-2+4)/2,(1+b)/2]
= [ 1 , (1+b)/2 ] ----( 1 )
Midpoint of BD =[(a+1)/2,(0+2)/2]
= [ (a+1)/2 , 1 ] ---( 2 )
Midpoint of AC= midpoint of BD
i ) ( a + 1 )/2 = 1
=> a + 1 = 2
=> a = 1
ii ) ( 1 + b )/2 = 1
=> 1 + b = 2
=> b = 1
Therefore ,
a = 1 , b = 1
iii ) length of the side AB
A(-2,1) = ( x1, y1)
B(1,0) = ( x2, y2 )
AB = √(x2-x1)² + (y2-y1)²
= √(1+2)²+(0-1)²
= √3²+1²
= √10
iv ) Length of the side BC
BC = √(4-1)²+(1-0)²
= √3²+1²
= √10
Therefore ,
AB = CD = √10
BC = AD = √10