Question

Show that A (−3, 2), B (−5, −5), C (2,−3), and D (4, 4) are the vertices of a rhombus.

 

Answer 1

Let A (−3, 2); B (−5,−5); C (2,−3) and D (4, 4) be the vertices of a quadrilateral. We have to prove that the quadrilateral ABCD is a rhombus.

So we should find the lengths of sides of quadrilateral ABCD.

`AB = sqrt((-5 + 3 )^2 + (-5-2)^2)`

       `= sqrt(4 + 4 9)`

       `= sqrt(53) `

`BC = sqrt((2 + 5 )^2 + (-3+ 5)^2)`

       `= sqrt(4 + 4 9)`

       `= sqrt(53) `

`CD = sqrt(( 4 - 2  )^2 + (4 +3)^2)`

       `= sqrt(4 + 4 9)`

       `= sqrt(53) `

`AD= sqrt((4 + 3 )^2 + (4-2)^2)`

       `= sqrt(4 + 4 9)`

       `= sqrt(53) `

All the sides of quadrilateral are equal. Hence ABCD is a rhombus.