Question

Show that the points A(6,1), B(8,2), C(9,4) and D(7,3) are the vertices of a rhombus. Find its area.

Answer 1

The given points are A(6,1), B(8,2), C(9,4) and D(7,3) .

`AB = sqrt ((6-8)^2 +(1-2)^2) = sqrt((-2)^2 +(-1)^2)`

`= sqrt(4+1) = sqrt(5) `

`BC = sqrt((8-9)^2 +(2-4)^2) = sqrt((-1)^2+(-2)^2)`

`= sqrt(1+4) = sqrt(5)`

`CD= sqrt((9-7) ^2 + (4-3)^2) = sqrt((2)^2 +(1)^2)`

`= sqrt(4+1) = sqrt(5)`

`AD = sqrt((7-6)^2 +(3-1)^2 ) = sqrt((1)^2 +(2)^2)`

`=sqrt (1+4) = sqrt(5)`

`AC = sqrt((6-9)^2 +(1-4)^2) = sqrt((-3)^2+(-3)^2)`

`= sqrt(9+9) = 3 sqrt(2)`

`=BD = sqrt(( 8-7)^2 +(2-3)^2) = sqrt((1)^2 +(-1)^2)`

`= sqrt(1+1) = sqrt(2)`

`∵ AB =BC = CD=AD = sqrt(5) and AC ≠ BD`

Therefore, the given points are the vertices of a rhombus. Now

Area` ( ΔABCD ) = 1/2 xx  AC xx BD`

` = 1/2 xx 3 sqrt(2) xx sqrt(2) = 3 ` sq. units

Hence, the area of the rhombus is 3 sq. units