If the points A(1,-2), B(2, 3,), C(a, 2) and D(-4,-3) form a parallelogram, then find the value of a and height of the parallelogram taking AB as base
Answers
Answer
Consider the given points A(1,−2),B(2,3),C(0,2) and D(−4,−3)
Since ABCD form a parallelogram, the midpoint of the diagonal AC should coincide with the midpoint of BD.
Mid point of AC= Mid point of BD
[
2
1+a
,
2
−2+2
]=[
2
2−4
,
2
3−3
]
[
2
a+1
,0]=[
2
−2
,0]
Since the mid points coincide, we have
2
1+a
=a
⇒a+1=−2
⇒a=−2−1
⇒a=−3
Now, area of ΔABC
=
2
1
∣x
1
(y
2
−y
3
)+x
2
(y
3
−y
1
)+x
3
(y
1
−y
2
)∣
=
2
1
∣1(3−2)+2(2−(−2))+(−3)(−2−3)∣
=
2
1
∣1(1)+2(4)+(−3)(−5)∣
=
2
1
∣1+8+15∣
=
2
24
=12 sq. units
ar(ABCD) parallelogram =2× Area of triangle
=2×12
=24 sq. units
Area of parallelogram =Base × Height
Base
Area
=height
So by the distance formula
=
(x
2
−x
1
)
2
+(y
2
−y
1
)
2
=
(−3+4)
2
+(2+3)
2
=
1+25
=
26
Thus height =
26
24
=
26
24
×
26
26
=
26
24
26
=
13
12
26
.: