5. If A(-2, I), 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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Step-by-step explanation:
11th
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If A( - 2, 1), B(a, 0), C(4...
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Asked on October 15, 2019 by
Arushi Chaulagain
If A(−2,1),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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ANSWER
Given sides of parallelogram A(−2,1),B(a,0),C(4,b),D(1,2)
We know that diagonals of Parallelogram bisect each other.
Mid-point let say O of diagonal AC is given by
x=(
2
x
1
+x
2
)and y=(
2
y
1
+y
2
)
O (
2
−2+4
,
2
1+b
) ..................(1)
Mid-point let say P of diagonal BD is given by
P (
2
a+1
,
2
0+2
) ..................(2)
Points O and P are same
Equating the corresponding co-ordinates of both midpoints, we get
2
−2+4
=
2
a+1
⇒a=1
and
2
1+b
=
2
0+2
⇒b=1
Now the Given co-ordinates of the parallelogram are written as
A(−2,1),B(1,0),C(4,1),D(1,2)
By distance formula,
(x
2
−x
1
)
2
+(y
2
−y
1
)
2
we can find the length of each side
AB=
(−2−1)
2
+(1−0)
2
AB=
(3)
2
+(1)
2
=
10
AB=CD ...............(pair of opposite sides of the parallelogram are parallel and equal)
BC=
(4−1)
2
+(1−0)
2
BC=
(3)
2
+(1)
2
=
10
BC=AD ...............(pair of opposite sides of the parallelogram are parallel and equal )
⇒AB=BC=CD=AD=
10
⇒ABCD is a Rhombus