Question

Verify that the following points taken in order to form the vertices of a rhombus

A(3, −2), B(7, 6), C(−1, 2) and D(−5, −6)

Answer 1

Distance = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`

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

= `sqrt((4)^2 + (8)^2`

= `sqrt(16 + 64)`

= `sqrt(80)`

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

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

= `sqrt(64 + 16)`

= `sqrt(80)`

CD = `sqrt((-5 + 1)^2 + (-6 - 2)^2`

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

= `sqrt(16 + 64)`

= `sqrt(80)`

AD = `sqrt((-5 - 3)^2 + (- 6 + 2)^2`

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

= `sqrt(64 + 16)`

= `sqrt(80)`

AB = BC = CD = AD = `sqrt(80)`

All the four sides are equal.

∴ ABCD is a rhombus.