(Q32)Show that the points (-4,-7),(-1,2),(8,5) and (5,-4) taken in order are the vertices of a rhombus. And find its area (hint: Area of rhombus =
1
2
 ×product of its diagonals) .

Answer

Given in ABCD A(-4,-7),B(-1,2),C(8,5) and D(5,-4)

Distance formula= √((x2 - x1)2 + (y2 - y1)2)

AB= √((-+)2 + (+)2)

= √

=√10

BC= √((+)2 + (-)2)

= √

= √10

CD= √((-)2 + (--)2)

= √

= √10

AD= √((+)2 + (-+)2)

= √

= √10

AC= √((+)2 + (+)2)

= √

=√2

BD= √((+)2 + (--)2)

= √

= √2

In ABCD AB=BC=CD=AD from sides are equal

Hence ABCD is a rhombus

Area of the rhombus=
1
2
 d1d2
=
1
2
 ×√2×√2

=sq.units.