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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.
Solution
`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.
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