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.
OR

If A(–5, 7), B(–4, –5), C(–1, –6) and D(4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD.

Question

Philoid · Accepted Answer

Given:
Parallelogram ABCD

A( &ndash; 2,1), B(a,0), C(4,b), D(4,5)

Now, Midpoint of AC = Midpoint of BD

(∵ diagonals of a parallelogram bisect each other)

Using Midpoint Formula i.e.

The midpoint of the segment joining  and  is given by

Putting values,

The midpoint of AC

The midpoint of BD

and

&rArr; a = &ndash; 2 and b = 4

Using Distance Formula i.e.

Distance between (x1, y1) and (x2, y2 ) is given by:

D =

Putting Values,

AB = &radic;(0 + 1) = 1 = CD

AD = &radic;(36 + 16) = &radic;52 = 2&radic;13 = BC

So,

AB = CD = 1; AD = BC = 2&radic;13

OR

Ar(ABCD) = Ar(ABC) + Ar(ADC)

The area of triangle with points (x1 , y1), (x2 , y2) and (x3 , y3) is:

&bull; Ar(ABC)

&bull; Ar(ADC)

So, Ar(ABCD)

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.ORIf A(–5, 7), B(–4, –5), C(–1, –6) and D(4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD.