Question

Show that `square` ABCD formed by the vertices A(-4,-7), B(-1,2), C(8,5) and D(5,-4) is a rhombus.

Answer 1

`AD = sqrt((X_2 -X_1)^2 +(Y_2 -Y_1)^2)`

`=sqrt(((5+4)^2 +(-4+7)^2`

`=sqrt(81+9)`

=`sqrt90`

=`3sqrt10              ................. (1)`

BC =`sqrt((8+1)^2 + (5-2)^2)`

`=sqrt(81+9)`

=`sqrt90`

`=3sqrt10                ............(2)`

`AB = sqrt(-1+4)^2 +(2+7)^2`

`=sqrt9+81`

`=sqrt90`

`=3sqrt10`      ...............(3)

`CD = sqrt((8-5)^2 +(5+4)^2)`

`=sqrt(9+81)`

`=sqrt90`

`=3sqrt10 `      ...............(4)

From (1), (2), (3) and (4); AB = BC = CD = DA
∴ `square` ABCD is a rhombus.