Question

Show that the following points are the vertices of a rectangle:

A(0, –4), B(6, 2), C(3, 5) and D(–3, –1)

Answer 1

The given points are A (0,-4), B(6,2), C(3,5) and D(-3,-1).

`AB = sqrt((6-0)^2 +{2-(-4)}^2) = sqrt((6)^2 +(6)^2) = sqrt(36+36) = sqrt(72) = 6 sqrt(2)  units`

`BC = sqrt(( 3-6)^2 + (5-2)^2) = sqrt((-3)^2 +(3)^2) = sqrt(9+9) = sqrt(18) = 3 sqrt(2)  units`

` CD = sqrt((-3-3)^2 +(-1-5)^2) = sqrt((-6)^2 +(-6)^2) = sqrt(36+36) = sqrt(72) = 6 sqrt(2)  units`

` AD = sqrt((-3-0)^2 + { -1-(-4)}^2) = sqrt((-3)^2 +(3)^2) = sqrt(9+9) = sqrt(18) = 3 sqrt(2)  units`

` Thus , AB =CD = sqrt(10) " units  and " BC = AD = sqrt(5)  units`

`Also ,  AC = sqrt((3-0)^2 +{ 5-(-4)}^2) = sqrt((3)^2 +(9)^2 )= sqrt(9+81) = sqrt(90) = 3 sqrt(10)  units`

`BD = sqrt((-3-6)^2 +(-1-2)^2) = sqrt((-9)^2 +(-3)^2) = sqrt(81+9) = sqrt(90) = 3 sqrt(10)  units`

Also, diagonal AC = diagonal BD

Hence, the given points from a rectangle